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Let $\textrm{Mat}_2(\mathbb{R})$ be the set of $2 \times 2$ matrices with real entries. For any $\varepsilon>0$ and any finitely--supported probability measure $\mu$ on $\textrm{Mat}_2(\mathbb{R})$, we prove that either \[ T(\mu) = \sum_{X, Y \in {\rm supp}(\mu), XY = YX} \mu(X) \mu(Y) < \varepsilon \] or there exists some finite set $S$ contained in a $2$-dimensional subspace of $\textrm{Mat}_2(\mathbb{R})$ such that $\mu(S) \gg \varepsilon^{3}$. We prove quantitatively stronger results when \[ \mu ( (a_{i,j})_{1\leq i,j \leq 2}) = \nu(a_{1,1}) \dots \nu(a_{2,2}) \ \ \text{for every} \ a_{1,1}, \dots, a_{2,2} \in \mathbb{R}, \] with $\nu$ being some finitely--supported probability measure on $\mathbb{R}$. For instance, when $A \subset \mathbb{R}$ is a proper generalised arithmetic progression of dimension $d$, and $\nu = 1_{A}/|A|$, our methods imply that $T(\mu) \ll_d |A|^{-3}$. This is optimal up to the $O_d(1)$ factor. Our proofs involve incidence geometric techniques as well as ideas surrounding the recent progress on Bourgain--Chang type sum-product estimates over $\mathbb{R}$. The latter includes Schmidt's subspace theorem and the resolution of the weak polynomial Freiman--Ruzsa conjecture over integers., Comment: 17 pages

A polytrope is a tropical polyhedron that is also classically convex. We study the tropical combinatorial types of polytropes associated to weighted directed acyclic graphs (DAGs). This family of polytropes arises in algebraic statistics when describing the model class of max-linear Bayesian networks. We show how the edge weights of a network directly relate to the facet structure of the corresponding polytrope. We also give a classification of polytropes from weighted DAGs at different levels of equivalence. These results give insight on the statistical problem of identifiability for a max-linear Bayesian network., Comment: 22 pages, 8 figures, 1 table

We provide lower bounds on the connectivity of the independence complexes of hypergraphs. Additionally, we compute the homotopy types of the independence complexes of $d$-uniform properly-connected triangulated hypergraphs., Comment: 15 pages

We construct a family of polytopes, which we call Chevalley polytopes, associated to homogeneous spaces $X=G/P$ in their projective embeddings $X\hookrightarrow \mathbb{P}(V_{\varpi})$ together with a choice of reduced expression for the minimal coset representative $w^P$ of $w_0$ in $W/W_P$. When $X$ is minuscule in its minimal embedding, we describe our construction in terms of order polytopes of minuscule posets and use the associated combinatorics to show that minuscule Chevalley polytopes are Newton-Okounkov bodies for $X$ and that the Pl\"ucker coordinates on $X$ form a Khovanskii basis for $\mathbb{C}[X]$. We conjecture similar properties for general $X$ and general embeddings $X\hookrightarrow\mathbb{P}(V_\varpi)$, along with a remarkable decomposition property which we consider as a polytopal shadow of the Littlewood-Richardson rule. We highlight a connection between Chevalley polytopes and string polytopes and give examples where Chevalley polytopes possess better combinatorial properties than string polytopes. We conclude with several examples further illustrating and supporting our conjectures., Comment: 24 pages

For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gr\"obner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach., Comment: 33 pages, 3 figures

Can outreach inspire and lead to research and vice versa? In this work, we introduce our approach to the gamification of research in mathematics and computer science through three illustrative examples. We discuss our primary motivations and provide insights into what makes our proposed gamification effective for three research topics in discrete and computational geometry and topology: (1) DominatriX, an art gallery problem involving polyominoes with rooks and queens; (2) Cubical Sliding Puzzles, an exploration of the discrete configuration spaces of sliding puzzles on the $d$-cube with topological obstructions; and (3) The Fence Challenge, a participatory isoperimetric problem based on polyforms. Additionally, we report on the collaborative development of the game Le Carr\'e du Diable, inspired by The Fence Challenge and created during the workshop Let's talk about outreach!, held in October 2022 in Les Diablerets, Switzerland. All of our outreach encounters and creations are designed and curated with an inclusive culture and a strong commitment to welcoming the most diverse audience possible., Comment: 16 pages, 11 figures

We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.

A strong $s$-blocking set in a projective space is a set of points that intersects each codimension-$s$ subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $(k - 1)$-dimensional projective space over $\mathbb{F}_q$ of size $O_s(q^s k)$, which is optimal up to the constant factor depending on $s$. This also yields an optimal explicit construction of affine blocking sets in $\mathbb{F}_q^k$ with respect to codimension-$(s+1)$ affine subspaces, and of $s$-minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong $1$-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong $s$-blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest., Comment: 17 pages, 4 figures, comments are welcome

The study of SAT and its variants has provided numerous NP-complete problems, from which most NP-hardness results were derived. Due to the NP-hardness of SAT, adding constraints to either specify a more precise NP-complete problem or to obtain a tractable one helps better understand the complexity class of several problems. In 1984, Tovey proved that bounded-degree SAT is also NP-complete, thereby providing a tool for performing NP-hardness reductions even with bounded parameters, when the size of the reduction gadget is a function of the variable degree. In this work, we initiate a similar study for QBF, the quantified version of SAT. We prove that, like SAT, the truth value of a maximum degree two quantified formula is polynomial-time computable. However, surprisingly, while the truth value of a 3-regular 3-SAT formula can be decided in polynomial time, it is PSPACE-complete for a 3-regular QBF formula. A direct consequence of these results is that Avoider-Enforcer and Client-Waiter positional games are PSPACE-complete when restricted to bounded-degree hypergraphs. To complete the study, we also show that Maker-Breaker and Maker-Maker positional games are PSPACE-complete for bounded-degree hypergraphs.

Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current state of the art and related bibliography) presented by participants of the 32nd edition of the workshop which took place in Poprad, Slovakia during September 8-13, 2024 (see the workshop webpage https://candc.upjs.sk)., Comment: Problems presented during the problem sessions of the 32nd Workshop on Cycles and Colours (Poprad, Slovakia, 8-13 September 2024)

Let $\lambda_{i}(G)$ be the $i$-th largest Laplacian eigenvalues of graph $G$, where $1\le i\le |V(G)|$. Liu, Yuan, You and Chen [Discrete Math., 341 (2018) 2969--2976] raised the problem for ``Which cospectral graphs have same degree sequences". In this paper, let $W_3$ and $W_5$ be the two graphs as shown in Fig. 2 and let $G$ be a connected graph with $n\ge 18$ vertices. We shall show that: $(1)$ If $\lambda_{2}(G)<5

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope $\triangle_\mathbb{H}$ (the Minkowski sum of the standard simplices $\triangle_H$ for all $H \in \mathbb{H}$). Hypergraphic posets include the weak order for the permutahedron (when $\mathbb{H}$ is the complete graph on $[n]$) and the Tamari lattice for the associahedron (when $\mathbb{H}$ is the set of all intervals of $[n]$), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of $[n]$. We characterize the interval hypergraphs $\mathbb{I}$ for which $P_\mathbb{I}$ is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order., Comment: 30 pages, 8 figures

Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrence of $w$ in the binary expansion of $n$ as a scattered subsequence. We study the behavior of the partial sum $\sum_{n=0}^N(-1)^{s_w(n)}$ and characterize several classes of words $w$ satisfying $\sum_{n=0}^N(-1)^{s_w(n)}= O(N^{1-\epsilon})$ for some $\epsilon >0$.

A recent result by Kardo\v{s}, M\'a\v{c}ajov\'a and Zerafa [J. Comb. Theory, Ser. B. 160 (2023) 1--14] related to the famous Berge-Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say $\mathcal O$, in a bridgeless cubic graph, there exists a $1$-factor intersecting all cycles in $\mathcal O$ in at least one edge. This remarkable result opens up natural generalizations in the case of an $r$-regular graph $G$ and a $t$-factor $F$, with $r$ and $t$ being positive integers. In this paper, we start the study of this problem by proving necessary and sufficient conditions on $G$, $t$ and $r$ to assure the existence of a suitable $F$ for any possible choice of the set $\mathcal O$. First of all, we show that $G$ needs to be $2$-connected. Under this additional assumption, we highlight how the ratio $\frac{t}{r}$ seems to play a crucial role in assuring the existence of a $t$-factor $F$ with the required properties by proving that $\frac{t}{r} \geq \frac{1}{3}$ is a further necessary condition. We suspect that this condition is also sufficient, and we confirm it in the case $\frac{t}{r}=\frac{1}{3}$, generalizing the case $t=1$ and $r=3$ proved by Kardo\v{s}, M\'a\v{c}ajov\'a, Zerafa, and in the case $\frac{t}{r}=\frac{1}{2}$ with $t$ even. Finally, we provide further results in the case of cycles of arbitrary length., Comment: 17 pages

We prove that there exist four operations such that given any two strongly $2$-connected digraphs $H$ and $D$ where $H$ is a butterfly-minor of $D$, there exists a sequence $D_0,\dots, D_n$ where $D_0=H$, $D_n=D$ and for every $0\leq i\leq n-1$, $D_i$ is a strongly $2$-connected butterfly-minor of $D_{i+1}$ which is obtained by a single application of one of the four operations. As a consequence of this theorem, we obtain that every strongly $2$-connected digraph can be generated from a concise family of strongly $2$-connected digraphs by using these four operations., Comment: 42 pages

The exclusivity principle (E-principle) states that the sum of probabilities of pairwise exclusive events cannot exceed 1. Unlike other principles proposed to characterize quantum correlations, its intrinsically non-bipartite formulation enables its application in more general contextuality scenarios. Although equivalent to the no-signalling condition for any bipartite Bell scenario, this equivalence breaks down for two independent copies of the same scenario. Such violations of the E-principle due to multiple copies, known as its activation effect, was studied in [Nat Commun 4, 2263 (2013)] for the nonlocal extremal boxes of $(2,m,2)$, $(2,2,d)$, and $(3,2,2)$ Bell scenarios. The authors mapped the problem of exhibiting activation effects to finding certain cliques inside joint exclusivity graphs. In this work, we refine the joint exclusivity structure to be an edge-colored exclusivity multigraph. This allows us to draw a novel connection to Ramsey theory, which guarantees the existence of certain monochromatic subgraphs in sufficiently large edge-colored cliques, providing a powerful tool for ruling out E-principle violations. We then exploit this connection, drawing on Ramsey-theoretic results to study violations of the E-principle by multiple copies of the contextual extremal boxes of $n$-cycle scenarios, called $n$-cycle PR boxes, for the usual ($n=4$) PR box we show that the known E-principle violation of $5/4$ is the maximal achievable using two copies, and that this same upper bound applies to two copies of the KCBS ($n = 5$) PR box. We then prove that $n \geq 6$-cycle PR boxes, unlike the extremal boxes of the aforementioned Bell scenarios, do not exhibit activation effects with two or three copies. Finally, for any number of independent copies $k$, we establish a lower bound on $n$ above which $n$-cycle PR boxes do not exhibit activation effects with $k$ copies., Comment: 17 pages, 14 figures

A graph $G$ has the \emph{Perfect Matching Hamiltonian property} (or for short, $G$ is $PMH$) if, for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. In this note, we show that \emph{Prism graphs} $\cP_n$ are not $PMH$, except for the $Cube\ graph$, and indicate for which values of $n$ the \emph{Crossed Prism graphs} $\cCP_n$ are $PMH$., Comment: 19 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:2106.00513

Given any positive integer $r$, Nahm's problem is to determine all rational $r\times r$ positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums wherein the original Nahm sum is modular while its dual is not modular. This provides counterexamples to Zagier's conjecture., Comment: First version

In Cayley graphs on the additive group of a small vector space over GF$(q)$, $q=2,3$, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph $G$ implies the existence of a CR code with the same parameters in the corresponding Hamming graph that covers $G$. In such a way, we find several completely regular codes with new parameters in Hamming graphs over GF$(3)$. The most interesting findings are two new CR-$1$ (with covering radius~$1$) codes that are independent sets (such CR are equivalent to optimal orthogonal arrays attaining the Bierbrauer--Friedman bound) and one new CR-$2$. By recursive constructions, every knew CR code induces an infinite sequence of CR codes (in particular, optimal orthogonal arrays if the original code was CR-$1$ and independent). In between, we classify feasible parameters of CR codes in several strongly regular graphs.

The concept of Vapnik-Chervonenkis (VC) density is pivotal across various mathematical fields, including model theory, discrete geometry, and probability theory. In this paper, we introduce a topological generalization of VC-density. Let $Y$ be a topological space and $\mathcal{X},\mathcal{Z}^{(0)},\ldots,\mathcal{Z}^{(q-1)}$ be families of subspaces of $Y$. We define a two parameter family of numbers, $\mathrm{vcd}^{p,q}_{\mathcal{X},\overline{\mathcal{Z}}}$, which we refer to as the degree $p$, order $q$, VC-density of the pair \[ (\mathcal{X},\overline{\mathcal{Z}} = (\mathcal{Z}^{(0)},\ldots,\mathcal{Z}^{(q-1)}). \] The classical notion of VC-density within this topological framework can be recovered by setting $p=0, q=1$. For $p=0, q > 0$, we recover Shelah's notion of higher-order VC-density for $q$-dependent families. Our definition introduces a new notion when $p>0$. Our main result establishes that that in any model of these theories \[ \mathrm{vcd}^{p,q}_{\mathcal{X},\overline{\mathcal{Z}}} \leq (p+q) \dim X. \] This result generalizes known VC-density bounds in these structures, extending them in multiple ways, as well as providing a uniform proof paradigm applicable to all of them. We give examples to show that our bounds are optimal. We present combinatorial applications of our higher-degree VC-density bounds, deriving topological analogs of well-known results such as the existence of $\varepsilon$-nets and the fractional Helly theorem. We show that with certain restrictions, these results extend to our higher-degree topological setting., Comment: 52 pages. Comments welcome

Benjamini, Yadin, and Yehudayoff (2007) showed that if the maximum degree of a graph $G$ is 'sub-logarithmic,' then the typical range of random $\mathbb Z$-homomorphisms is super-constant. Furthermore, they showed that there is a sharp transition on the range of random $\mathbb Z$-homomorphisms on the graph $C_{n,k}$, the tensor product of the $n$-cycle and the complete graph on $k$ vertices with self-loops, around $k=2\log n$. We extend (to some extent) their results to random $M$-Lipschitz functions and random real-valued Lipschitz functions., Comment: 16 pages, 1 figure

Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if $\mathbf{a} \in \mathbb{R}^{\mathcal{H}}$ is such that the weighted arrangement $(\mathcal{H}, \mathbf{a})$ is \emph{stable}, then $Q(\mathbf{a}) \leq 0$. As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a, \ldots, a) \leq 0$ gives a lower bound for the total sum of multiplicities of codimension $2$ intersection subspaces of $\mathcal{H}$. The lower bound is attained when every $H \in \mathcal{H}$ intersects all the other members of $\mathcal{H} \setminus \{H\}$ along $(1-2/(n+1))|\mathcal{H}| + 1$ codimension $2$ subspaces; extending from $n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane., Comment: 117 pages

We develop novel tools for computing the likelihood correspondence of an arrangement of hypersurfaces in a projective space. This uses the module of logarithmic derivations. This object is well-studied in the linear case, when the hypersurfaces are hyperplanes. We here focus on nonlinear scenarios and their applications in statistics and physics., Comment: 20 pages, 1 figure

Path homology plays a central role in digraph topology and GLMY theory more general. Unfortunately, the computation of the path homology of a digraph $G$ is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In this paper we introduce an inductive method of constructing elements of the path homology chain modules $\Omega_n(G;R)$ from elements in the proceeding two dimensions. This proceeds via the formation of what we call upper and lower \emph{extensions}, that are parametrised by certain labeled multihypergraphs which we introduce and call \emph{face multihypergraphs}. When the coefficient ring $R$ is a finite field the inductive elements we construct generate $\Omega_*(G;R)$. With integral or rational coefficients, the inductive elements generate at least $\Omega_i(G;R)$ for $i=0,1,2,3$. Since in low dimensions the inductive elements extended over labeled multigraphs coincide with naturally occurring generating sets up to sign, they are excellent candidates to reduce to a basis. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph $G$. We employ inductive elements to construct a sequence of digraphs whose path Euler characteristic can differ arbitrarily depending on the choice of field coefficients. In particular, answering an open question posed by Fu and Ivanov.

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $\{1,2,\ldots,n\}$. Let $t, s, k, n$ be non-negative integers with $k \geq s+1$ and $n \geq 2 k+t$. In 2016, Frankl proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is $(t+1)$-intersecting and $|\mathcal{F}| \geq 1$, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Furthermore, Frankl conjectured that under an additional condition $\binom{[k+t+s]} {k+t}\subseteq\mathcal{F}$, the following inequality holds: $$ |\mathcal{F}|+|\mathcal{G}| \leq\binom{k+t+s}{k+t}+\binom{n}{k}-\sum_{i=0}^s\binom{k+t+s}{i}\binom{n-k-t-s}{k-i}. $$ In this paper, we prove this conjecture. The key ingredient is to establish a theorem for cross-intersecting families with a restricted universe. Moreover, we derive an analogous result for this conjecture., Comment: 10 pages. arXiv admin note: text overlap with arXiv:2411.08546, arXiv:2411.03674

Our main result is a combinatorial characterization of when a horospherical variety has (at worst) quotient singularities. Using this characterization, we show that every quasiprojective horospherical variety with quotient singularities is globally the quotient of a smooth variety by a finite abelian group., Comment: 11 pages; comments welcome

For a graph $G$, its $k$-th power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$ of each other. The $k$-independence number $\alpha_k(G)$ is defined as the independence number of $G^k$. By using general semidefinite programming and polynomial methods, we derive sharp bounds for the $k$-independence number of a graph, which extend and unify various existing results. Our work also allows us to easily derive some new bounds for $\alpha_k(G)$.

How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $\gamma_d$. It is easy to see that $\gamma_d \leq 1/(d+1)$, and it is known that $\gamma_d=1/(d+1)$ for $d \leq 2$, but it was recently shown that $\gamma_d < 1/(d+1)$ for $d \geq 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$., Comment: 8 pages

In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in order to provide a probabilistic formula for the number of bipartite graphs with fixed surplus. Using this, we obtain asymptotics as the number of vertices in each class tend to infinity., Comment: 15 pages, 2 figures

This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always sits between two ordinary matroids and use this fact to prove that it has many of the same properties of ordinary matroids. We compute the dimension of its order complex using its M\"obius function, We show that its matroid polytope is geometrically defined using its flats and connected to its Bergman fan. We finish by highlighting differences between its toric verity and the toric verity of an ordinary matroid, and give a partial proof of Mason's conjecture.

Ramsey's theorem states that if $N$ is sufficiently large, then no matter how one colors the edges among $N$ vertices with two colors, there are always $k$ vertices spanning edges in only one color. Given this theorem, it is natural to ask ``how large is sufficiently large?'' Ramsey's original proof showed that $N=k!$ is sufficient, and five years later Erd\H{o}s and Szekeres improved this bound to $N=4^k$. And then progress stalled for almost 90 years. In this survey, I present the history of the problem and discuss some of the ideas used in the recent breakthrough of Campos--Griffiths--Morris--Sahasrabudhe, who proved that $N=3.993^k$ is sufficient., Comment: Expository paper accompanying a Bourbaki seminar talk (November 2024, expos\'e number 1230). 33 pages

A $k$-star is a complete bipartite graph $K_{1,k}$. A partial $k$-star design of order $n$ is a pair $(V,\mathcal{A})$ where $V$ is a set of $n$ vertices and $\mathcal{A}$ is a set of edge-disjoint $k$-stars whose vertex sets are subsets of $V$. If each edge of the complete graph with vertex set $V$ is in some star in $\mathcal{A}$, then $(V,\mathcal{A})$ is a (complete) $k$-star design. We say that $(V,\mathcal{A})$ is completable if there is a $k$-star design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. In this paper we determine, for all $k$ and $n$, the minimum number of stars in an uncompletable partial $k$-star design of order $n$., Comment: 16 pages, 0 figures

Pandharipande-Thomas theory and Donaldson-Thomas theory (PT and DT) are two branches of enumerative geometry in which particular generating functions arise that count plane-partition-like objects. That these generating functions differ only by a factor of MacMahon's function was proven recursively by Jenne, Webb, and Young using the double dimer model. We bijectivize two special cases of the result by formulating these generating functions using vertex operators and applying a particular type of local involution known as a toggle, first introduced in the form we use by Pak., Comment: 31 pages, 29 figures

In this paper, we develop the concept of multiple cylinder of relations which is a generalization of the relation cylinder, extending the multiple non-Hausdorff mapping cylinder to sequences of finite T0-spaces linked by a series of relations. This construction is important in capturing complex homotopical structures across chains of finite spaces and, when the relations are induced by maps, it serves as a third space that collapses to two distinct finite spaces. Additionally, we introduce the concept of a strong-good cover for simplicial complexes and finite spaces, char acterized by collapsible (rather than merely contractible) intersections. This leads to a strengthened version of the Nerve Theorem, which we develop for simplicial complexes as well as for finite spaces with strong-good covers, demonstrating that these complexes and spaces and their associated nerves maintain the same simple homotopy type, thereby refining classical results for finite simplicial complexes and finite topological structures.

We study higher order quantum maps in the context of a *-autonomous category of affine subspaces. We show that types of higher order maps can be identified with certain Boolean functions that we call type functions. By an extension of this identification, the algebraic structure of Boolean functions is inherited by some sets of quantum objects including higher order maps. Using the M\"obius transform, we assign to each type function a poset whose elements are labelled by subsets of indices of the involved spaces. We then show that the type function corresponds to a comb type if and only if the poset is a chain. We also devise a procedure for decomposition of the poset to a set of basic chains from which the type function is constructed by taking maxima and minima of concatenations of the basic chains in different orders. On the level of higher order maps, maxima and minima correspond to affine mixtures and intersections, respectively., Comment: 51 pages, figures. Comments welcome

Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $\operatorname {sym}(n)$, for $3\leq n \leq 6$, have been studied and completely determined. The case for $\operatorname {sym}(7)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\operatorname {sym}(7)$ with a minimal distance of at least $4$. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $\operatorname {sym}(7)$ are determined.

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbf{\Sigma}^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs., Comment: 15 pages

We consider equilibrium one-on-one conversations between neighbors on a circular table, with the goal of assessing the likelihood of a (perhaps) familiar situation: sitting at a table where both of your neighbors are talking to someone else. When $n$ people in a circle randomly prefer their left or right neighbor, we show that the probability a given person is unmatched in equilibrium (i.e., in a stable matching) is $$\frac{1}{9} + \left(\frac{1}{2}\right)^n\left(\frac{2n}{3} - \frac{8}{9} + \frac{2}{n}\right)$$ for odd $n$ and $$\frac{1}{9} - \left(\frac{1}{2}\right)^n\left(\frac{2n}{3} - \frac{8}{9}\right)$$ for even $n$. This probability approaches $1/9$ as $n\rightarrow \infty$. We also show that the probability \textit{every} person is matched in equilibrium is $0$ for odd $n$ and $\frac{3^{n/2}-1}{2^{n-1}}$ for even $n$.

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete bipartite graph $K_{t,t}$ as a subgraph. In this case we show that $G$ has an induced path of order $(\log \log n)^{1/5-o(1)}$. This is an exponential improvement over a result of Galvin, Rival, and Sands (1982) and comes close to a recent upper bound of order $O((\log \log n)^2)$. Another way to approach this problem is by viewing $G$ as an ordered graph (where the vertices are ordered according to their position on the path $P$). From this point of view it is most natural to consider which ordered subgraphs need to be forbidden in order to force the existence of a long induced path. Focusing on the exclusion of ordered matchings, we improve or recover a number of existing results with much simpler proofs, in a unified way. We also show that if some forbidden ordered subgraph forces the existence of a long induced path in $G$, then this induced path has size at least $\Omega((\log \log \log n)^{1/3})$, and can be chosen to be increasing with respect to $P$., Comment: 26 pages, 8 figures. Comments welcome !

A combinatorial game is a two-player game without hidden information or chance elements. One of the major approaches to analyzing games in combinatorial game theory is to break down a given game position into a disjunctive sum of multiple sub-positions, then evaluate the game value of each component of the sum, and finally integrate these game values to find which player has a winning strategy in the whole position. Accordingly, finding the game value of a given position is a major topic in combinatorial game theory. The sequential compound proposed by Stromquist and Ullman is a combinatorial game consisting of two combinatorial games. In the sequential compound of games $G$ and $H$, the players make moves on $G$ until $G$ is over, and then they play on $H$. In this paper, we investigate the general properties of sequential compounds. As the main result, we give the game values of sequential compounds of a finite number of integers and stars, which are basic and typical games in combinatorial game theory.

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for $k \geq 1, t\ge 0$ and $n \geq 2 k+t$, if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, in which $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. This bound can be attained when $\mathcal{F}$ consists of a single set. In this paper, we generalize this result under the constraint $|\mathcal{F}| \geq r$ for every $r\leq n-k-t+1$. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the $s$-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results., Comment: 17pages. arXiv admin note: text overlap with arXiv:2411.03674

An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n, \mathcal{G})$ is the smallest number of colors $r$ required to ensure that, no matter how the edges of the complete graph $K_n$ are colored using exactly $r$ colors, there will always be a rainbow copy of some graph $G$ from the family $\mathcal{G}$. A friendship graph $F_k$ is the graph obtained by combining $k$ triangles that share a common vertex. In this paper, we determine the anti-Ramsey number $ar(n, \{F_k\})$ for large values of $n$. Additionally, we also determine the $ar(n, \{K_{1,k}, M_k\}$, where $K_{1,k}$ is a star graph with $ k+1$ vertices and $M_k$ is a matching of size $k$.

The fact that Schubert polynomials are the weighted counting functions for reduced RC-graphs, also known as reduced pipe dreams, was established using their generating functions inside an appropriate Demazure algebra. Here we investigate the generating functions of another family of polynomials, the key polynomials, also known as Demazure characters. Each component in that function is a rational function, whose denominator is an explicit product whose definition is based on bounded ascending sequences of integers. We determine the first terms of the polynomial numerator, and pose conjectures about these terms in general as well as some of the next ones. The form of our generating functions suggests relations between the coefficients in key polynomials and signed sums of numbers of integral points on polytopes., Comment: 48 pages

Pick $n$ independent and uniform random points $U_1,\ldots,U_n$ in a compact convex set $K$ of $\mathbb{R}^d$ with volume 1, and let $P^{(d)}_K(n)$ be the probability that these points are in convex position. The Sylvester conjecture in $\mathbb{R}^d$ is that $\min_K P^{(d)}_K(d+2)$ is achieved by the $d$-dimensional simplices $K$ (only). In this paper, we focus on a companion model, already studied in the $2d$ case, which we define in any dimension $d$: we say that $K$ has $F$ as a flat floor, if $F$ is a subset of $K$, contained in a hyperplan $P$, such that $K$ lies in one of the half-spaces defined by $P$. We define $Q_K^F(n)$ as the probability that $U_1,\cdots,U_n$ together with $F$ are in convex position (i.e., the $U_i$ are on the boundary of the convex hull ${\sf CH}(\{U_1,\cdots,U_n\}\cup F\})$). We prove that, for all fixed $F$, $K\mapsto Q_K^F(2)$ reaches its minimum on the "mountains" with floor $F$ (mountains are convex hull of $F$ union an additional vertex), while the maximum is not reached, but $K\mapsto Q_K^F(2)$ has values arbitrary close to 1. If the optimisation is done on the set of $K$ contained in $F\times[0,d]$ (the "subprism case"), then the minimum is also reached by the mountains, and the maximum by the "prism" $F\times[0,1]$. Since again, $Q_K^F{(2)}$ relies on the expected volume (of ${\sf CH}(\{V_1,V_2\}\cup F\})$), this result can be seen as a proof of the Sylvester problem in the floor case. In $2d$, where $F$ can essentially be the segment $[0,1],$ we give a general decomposition formula for $Q_K^F(n)$ so to compute several formulas and bounds for different $K$. In 3D, we give some bounds for $Q_K^F(n)$ for various floors $F$ and special cases of $K$., Comment: 28 pages, 8 figures

Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations. Using combinatorial species and sign-reversing involutions we derive counting formulas and generating functions for the Caylerian polynomials as well as for related refined polynomials., Comment: 23 pages, 3 tables

We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/k\rceil$ times. Call a graph equitably $k$-choosable if it has an equitable $L$-coloring for every $k$-list assignment $L$. A graph $G$ is $(a,b)$-sparse if for every $A\subseteq V(G)$, the number of edges in the subgraph $G[A]$ of $G$ induced by $A$ is at most $a|A|+b$. Our first main result is that every $(\frac{7}{6},\frac{1}{3})$-sparse graph with minimum degree at least $2$ is equitably $3$-colorable and equitably $3$-choosable. This is sharp. Our second main result is that every $(\frac{5}{4},\frac{1}{2})$-sparse graph with minimum degree at least $2$ is equitably $4$-colorable and equitably $4$-choosable. This is also sharp. One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.

A code ${\mathcal C}$ is a subset of the vertex set of a Hamming graph $H(n,q)$, and ${\mathcal C}$ is $2$-neighbour-transitive if the automorphism group $G={\rm Aut}({\mathcal C})$ acts transitively on each of the sets ${\mathcal C}$, ${\mathcal C}_1$ and ${\mathcal C}_2$, where ${\mathcal C}_1$ and ${\mathcal C}_2$ are the (non-empty) sets of vertices that are distances $1$ and $2$, respectively, (but no closer) to some element of ${\mathcal C}$. Suppose that ${\mathcal C}$ is a $2$-neighbour-transitive code with minimum distance at least $5$. For $q=2$, all `minimal' such ${\mathcal C}$ have been classified. Moreover, it has previously been shown that a subgroup of the automorphism group of the code induces an affine $2$-transitive group action on the alphabet of the Hamming graph. The main results of this paper are to show that this affine $2$-transitive group must be a subgroup of ${\rm A}\Gamma{\rm L}_1(q)$ and to provide a number of infinite families of examples of such codes. These examples are described via polynomial algebras related to representations of certain classical groups., Comment: 18 pages

A graph is said to be $F$-free if it does not contain $F$ as a subgraph. Brualdi-Hoffman-Tur\'{a}n problem seeks to determine the maximum spectral radius of an $F$-free graph with given size. The gem consists of a path on $4$ vertices, along with an additional vertex that is adjacent to every vertex of the path. Concerning Brualdi-Hoffman-Tur\'{a}n problem of the gem, when the size is odd, Zhang and Wang [Discrete Math. 347 (2024) 114171] and Yu, Li and Peng [arXiv:2404. 03423] solved it. In this paper, we completely solve the Brualdi-Hoffman-Tur\'{a}n problem type problem of the gem.

We prove that, for every set of $n$ points $\mathcal{P}$ in $\mathbb{R}^2$, a random plane graph drawn on $\mathcal{P}$ is expected to contain less than $n/10.18$ isolated vertices. In the other direction, we construct a point set where the expected number of isolated vertices in a random plane graph is about $n/23.32$. For $i\ge 1$, we prove that the expected number of vertices of degree $i$ is always less than $n/\sqrt{\pi i}$ Our analysis is based on cross-graph charging schemes. That is, we move charge between vertices from different plane graphs of the same point set. This leads to information about the expected behavior of a random plane graph.