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In this paper, we present an algorithm on vanishing coefficients with arithmetic progressions in the sum of two generalized eta-quotients by using the theory of modular forms, and utilize our algorithm to prove five conjectures on matching coefficients in series expansions of certain q -products and their reciprocals. One of which was provided by Baruah and Das, and the others were found by Schlosser. For instance, we prove that for any n ≥ 0 , λ 12 (10 n + r) = λ 12 ′ (10 n + r − 6) , r ∈ { 7 , 9 } , where the sequences { λ 12 (n) } n ≥ 0 and { λ 12 ′ (n) } n ≥ 0 are defined by ∑ n = 0 ∞ λ 12 (n) q n = 1 R (q) R (q 2) R (q 4) R (q 8) = (∑ n = 0 ∞ λ 12 ′ (n) q n) − 1 , and where R (q) is the Rogers–Ramanujan continued fraction, which has the following celebrated product representation: R (q) = ∏ n = 0 ∞ (1 − q 5 n + 1) (1 − q 5 n + 4) (1 − q 5 n + 2) (1 − q 5 n + 3). Finally, we find that this phenomenon also exists in other infinite q -series expansions. [ABSTRACT FROM AUTHOR]

Let G be a bridgeless cubic graph. The Berge–Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan–Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that G admits two perfect matchings whose deletion yields a bipartite subgraph of G. It can be shown that given an arbitrary perfect matching of G , it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan–Raspaud and the Berge–Fulkerson conjectures, respectively. In this paper, we show that given any 1 + -factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G , there always exists a perfect matching M of G containing e such that G ∖ (F ∪ M) is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in G , there exists a perfect matching of G containing at least one edge of each circuit in this collection. [ABSTRACT FROM AUTHOR]

Let G be a simple graph, and let χ ′ (G) and Δ (G) denote the chromatic index and the maximum degree of G , respectively. A graph G is a critical class two graph if χ ′ (G) = Δ (G) + 1 and χ ′ (H) ≤ Δ (G) for every proper subgraph H of G. Let d ‾ (G) denote the average degree of G , i.e., d ‾ (G) = 2 | E (G) | / | V (G) |. Vizing in 1968 conjectured that d ‾ (G) ≥ Δ (G) − 1 + 3 / n if G is a critical class two graph of order n. In this paper, we prove that d ‾ (G) ≥ 3 4 Δ (G) − 8 if G is a critical class two graph. Let δ (G) denote the minimum degree of G. We show that there exist two functions D and D 0 such that for any ϵ ∈ (0 , 1) , if G is a critical class two graph with Δ (G) ≥ D (ϵ) and δ (G) ≥ D 0 (ϵ) , then d ‾ (G) ≥ (1 − ϵ) Δ (G). We will give two specific functions satisfying the statement above in the paper. Moreover, we show that if G is a critical class two graph and δ (G) ≥ (log ⁡ Δ (G)) 3 4 , then d ‾ (G) ≥ Δ (G) − o (Δ (G)). [ABSTRACT FROM AUTHOR]

In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers n , k , r , s with n ≥ r k and 2 ≤ s ≤ r , the r -uniform general Kneser hypergraph KG s r (n , k) , has all k -subsets of { 1 , ... , n } as the vertex set and all multi-sets { A 1 , ... , A r } of k -subsets with s -wise empty intersections as the edge set. The case r = s = 2 , was considered by Kneser [7] in 1955, where he conjectured that its chromatic number is n − 2 (k − 1). This was finally proved by Lovász [9] in 1978. The case r > 2 and s = 2 , was considered by Erdös in 1973, and he conjectured that its chromatic number is ⌈ n − r (k − 1) r − 1 ⌉. This conjecture was proved by Alon, Frankl and Lovász [2] in 1986. The case where s > 2 , was considered by Sarkaria [11] in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler [14] in 2006 in the induction method of Sarkaria on the number of prime factors of r , and Sarkaria's proof only worked when s is less than the smallest prime factor of r or s = 2. Finally in 2019, Aslam, Chen, Coldren, Frick and Setiabrata [3] were able to extend this by using methods different from Sarkaria to the case when r = 2 α 0 p 1 α 1 ... p t α t and 2 ≤ s ≤ 2 α 0 (p 1 − 1) α 1 ... (p t − 1) α t . In this paper, by applying the Z p -Tucker lemma of Ziegler [13] and Meunier [10] , we finally prove the general Erdös conjecture and prove the claimed result of Sarkaria for any 2 ≤ s ≤ r. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lovász [2] and compute the connectivity of certain simplicial complexes that might be of interest in their own right. [ABSTRACT FROM AUTHOR]

Let Γ be a connected (X , 2) -arc-transitive bipartite graph with bi-parts Δ 0 and Δ 1 , where X ≤ Aut (Γ). Let X + be the subgroup of X fixing Δ 0 setwise. In this paper, we first prove that if X + is primitive and faithful on Δ 0 and Δ 1 , then the actions of X + on Δ 0 and Δ 1 are primitive of the same type X with X ∈ { HA , SA , PA }. This is then used to prove that if X + is quasiprimitive on Δ 0 of type HA or TW, then either soc (X +) ⊴ Aut (Γ) , or Γ ≅ K q , q (q a prime power) or Γ ≅ K 2 r , 2 r − 2 r K 2 (r ≥ 2). This confirms a conjecture of C. H. Li from 2008. [ABSTRACT FROM AUTHOR]

Let G = O p ′ ( G ¯ F) be a finite simple group of Lie type defined over a field of characteristic p , where F is a Steinberg endomorphism of the ambient simple algebraic group G ¯. Let T ¯ be an F -stable maximal torus of G ¯ and set N = N G (T ¯). A conjecture due to Vdovin asserts that if G ≇ L 3 (2) then N ∩ N x is a p -group for some x ∈ G. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of G on G / N , which allows us to prove a stronger, and suitably modified, version of Vdovin's conjecture. [ABSTRACT FROM AUTHOR]

Let 1 ⩽ a < b be two relatively prime integers. Sylvester found that a b − a − b is the largest integer which can not be represented by a x + b y (x , y ∈ Z ⩾ 0) about 160 years ago and this number shall be denoted by g a , b. Let N (a , b) = { n : n ⩽ g a , b , n = a x + b y , x , y ∈ Z ⩾ 0 } and π a , b be the number of primes in N (a , b). Recently, Ramírez Alfonsín and Skałba proved that π a , b ≫ ε g a , b (log ⁡ g a , b) 2 + ε for any fixed ε > 0. They further conjectured that the order of the magnitude of π a , b is 1 2 π (g a , b) , where π (x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a , b with 1 ⩽ a < b and (a , b) = 1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

Let E / Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of E / Q divides the product of the order of the Shafarevich–Tate group of E / Q , the (global) Tamagawa number of E / Q , and the Tamagawa number of E / Q at infinity. This consequence of the Birch and Swinnerton-Dyer conjecture was noticed by Agashe and Stein in 2005. In this paper, we prove this divisibility statement unconditionally in many cases, including the case where the curve E / Q is semi-stable. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new invariants to a singularity (V , 0) , i.e., the derivation Lie algebras L k (V) of the higher Nash blow-up local algebra M k (V). A new conjecture about the non-existence of negative weighted derivations of L k (V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L 2 (V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of L k (V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number and k an imaginary quadratic field in which p splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for p and k , which states that the non-trivial Iwasawa module of the maximal multiple Z p -extension field over k has a non-trivial pseudo-null submodule. We prove this conjecture for p and k under the assumption that the Iwasawa λ -invariants vanish for the Z p -extensions over k in which one of the prime ideals of k lying above p do not ramify and that the characteristic ideal of the Iwasawa module associated to the cyclotomic Z p -extension over k has a square-free generator. [ABSTRACT FROM AUTHOR]

For a simple graph F , let Ex (n , F) and E x sp (n , F) denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an n -vertex graph without any copy of the graph F , respectively. The Turán graph T n , r is the complete r -partite graph on n vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let F be any graph such that the graphs in Ex (n , F) are Turán graphs plus O (1) edges. Then E x sp (n , F) ⊂ Ex (n , F) for sufficiently large n. In this paper we consider the graph F such that the graphs in Ex (n , F) are obtained from T n , r by adding O (1) edges, and prove that if G has the maximum spectral radius among all n -vertex graphs not containing F , then G is a member of Ex (n , F) for n large enough. Then Cioabă, Desai and Tait's conjecture is completely solved. Furthermore, we give a stronger result. [ABSTRACT FROM AUTHOR]

In this paper we investigate the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph G and the maximal number of distinct degrees appearing in an induced subgraph of G , denoted respectively by hom ⁡ (G) and f (G). Our main theorem improves estimates due to several earlier researchers and shows that if G is an n -vertex graph with hom ⁡ (G) ≥ n 1 / 2 then f (G) ≥ (n / hom ⁡ (G)) 1 − o (1). The bound here is sharp up to the o (1) -term, and asymptotically solves a conjecture of Narayanan and Tomon. In particular, this implies that max ⁡ { hom ⁡ (G) , f (G) } ≥ n 1 / 2 − o (1) for any n -vertex graph G , which is also sharp. The above relationship between hom ⁡ (G) and f (G) breaks down in the regime where hom ⁡ (G) < n 1 / 2. Our second result provides a sharp bound for distinct degrees in biased random graphs, i.e. on f (G (n , p)). We believe that the behaviour here determines the extremal relationship between hom ⁡ (G) and f (G) in this second regime. Our approach to lower bounding f (G) proceeds via a translation into an (almost) equivalent probabilistic problem, and it can be shown to be effective for arbitrary graphs. It may be of independent interest. [ABSTRACT FROM AUTHOR]

Let G be a K p -free graph. We say e is a K p -saturating edge of G if e ∉ E (G) and G + e contains a copy of K p. Denote by f p (n , m) the minimum number of K p -saturating edges that an n -vertex K p -free graph with m edges can have. Erdős and Tuza conjectured that f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) n 2 16. Balogh and Liu disproved this by showing f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) 2 n 2 33. They believed that a natural generalization of their construction for K p -free graph should also be optimal and made a conjecture that f p + 1 (n , ex (n , K p) + 1) = (2 (p − 2) 2 p (4 p 2 − 11 p + 8) + o (1)) n 2 for all integers p ≥ 3. The main result of this paper is to confirm the above conjecture of Balogh and Liu. [ABSTRACT FROM AUTHOR]

Lovász and Cherkassky discovered in the 1970s independently that if G is a finite graph with a given set T of terminal vertices such that G is inner Eulerian with respect to T , then the maximal number of edge-disjoint paths connecting distinct vertices in T is ∑ t ∈ T λ (t , T − t) where λ is the local edge-connectivity function. The optimality of a system of edge-disjoint T -paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous paper in Li and Zhang (2018) [13]. We prove that if (n , q − 1) = 1 , PSL n (q) satisfies the inductive blockwise Alperin weight condition for any prime ℓ different from the defining characteristic. [ABSTRACT FROM AUTHOR]

In this paper it is shown that there is an absolute constant κ > 0 with the following property. For any prime p and nonempty subsets A , B of Z / p Z such that 1 < | A | < p 2 , B ∩ − B = ∅ and | B | < κ | A | ln ⁡ (| A |) , we have that max b ∈ B ⁡ | (A + b) ∖ A | ≥ | B |. In 2011, V. Lev proved the former statement assuming also that | B | ≤ p 8 ; in the same paper, Lev suggested that this technical condition could be eliminated or weakened. In this paper his conjecture is confirmed. [ABSTRACT FROM AUTHOR]

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K 5. This conjecture was proved by Ma and Yu for graphs containing K 4 − , and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequent papers to prove the Kelmans-Seymour conjecture. [ABSTRACT FROM AUTHOR]

Let f be any arithmetic function and define S f (x) : = ∑ n ⩽ x f ([ x / n ]). When f equals the Euler totient function φ , several authors studied the upper and lower bounds of S φ (x). In this paper we shall prove that S φ (x) has an asymptotic formula by the method of exponential sums. This result proves a conjecture proposed by Bordellés, Dai, Heyman, Pan and Shparlinski. Some other asymptotic formulas for arbitrary f are also given in this paper. [ABSTRACT FROM AUTHOR]

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q ≥ max ⁡ { (2 k − 1) 2 + 1 , 1.823 (4 k 2 − 14 k + 12) } , whereas Conjecture B holds if q ≥ 2.233 (9 k 2 − 21 k + 12). Although one of these conjectures was already proved by Hou without any restriction on the degree of the polynomials, we consider the proof contained in this paper is more direct and less computational. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ 0 (N) with level N and nebentypus χ N. Let Q (x) be a positive definite integral quadratic form, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λ f (n) and r (n , Q). We generalize and improve previous results without the Ramanujan conjecture. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

χ -bounded classes are studied here in the context of star colorings and, more generally, χ p -colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ 2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χ p -boundedness leads to more stability and we give structural characterizations of (strong and weak) χ p -bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most φ (g , ω (G)) | G | holes of length g. [ABSTRACT FROM AUTHOR]

This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots of non-self-intersecting curves in an n -punctured disc and the set of real Schur roots of acyclic (valued) quivers with n vertices. [ABSTRACT FROM AUTHOR]

A graph G = (V , E) is total weight (k , k ′) -choosable if the following holds: For any list assignment L which assigns to each vertex v a set L (v) of k real numbers, and assigns to each edge e a set L (e) of k ′ real numbers, there is a proper L -total weighting, i.e., a map ϕ : V ∪ E → R such that ϕ (z) ∈ L (z) for z ∈ V ∪ E , and ∑ e ∈ E (u) ϕ (e) + ϕ (u) ≠ ∑ e ∈ E (v) ϕ (e) + ϕ (v) for every edge { u , v }. A graph is called nice if it contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture, it was conjectured in Wong and Zhu (2011) [23] that every nice graph is total weight (1 , 3) -choosable. The problem whether there is a constant k such that every nice graph is total weight (1 , k) -choosable remained open for a decade and was recently solved by Cao (2021) [6] , who proved that every nice graph is total weight (1 , 17) -choosable. This paper improves this result and proves that every nice graph is total weight (1 , 5) -choosable. [ABSTRACT FROM AUTHOR]

Minors play a key role in graph theory, and extremal problems on forbidding minors have attracted appreciable amount of interest in the past decades. In this paper, we focus on spectral extrema of K s , t -minor free graphs, and determine extremal graphs with maximum spectral radius over all K s , t -minor free graphs of sufficiently large order. This generalizes and improves several previous results. For t ≥ s ≥ 2 , our result completely solves Tait's conjecture. For t ≥ s = 1 , our result gives a spectral analogue of a theorem due to Ding, Johnson and Seymour, which determines the maximum number of edges in K 1 , t -minor free connected graphs. Some spectral and structural tools, such as, local edge maximality, local degree sequence majorization and double eigenvectors transformation, are used to characterize structural properties of extremal graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with maximum degree Δ (G). A subgraph H of G is overfull if | E (H) | > Δ (G) ⌊ | V (H) | / 2 ⌋. Chetwynd and Hilton in 1986 conjectured that a graph G on n vertices with Δ (G) > n / 3 has chromatic index Δ (G) if and only if G contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative. [ABSTRACT FROM AUTHOR]

The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every n -vertex graph with at least ε n 2 edges contains a 1-subdivision of the complete graph on c ε n vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least (2 + o (1)) k 2 vertices contains the 1-subdivision of a transitive tournament on k vertices. This is optimal up to a multiplicative factor of 4 and confirms a conjecture of Girão, Popielarz and Snyder. [ABSTRACT FROM AUTHOR]

Let ⊲ be a relation between graphs. We say a graph G is ⊲ -ubiquitous if whenever Γ is a graph with n G ⊲ Γ for all n ∈ N , then one also has ℵ 0 G ⊲ Γ , where αG is the disjoint union of α many copies of G. The Ubiquity Conjecture of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979. [ABSTRACT FROM AUTHOR]

Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size q with a subspace structure in pseudo-Paley graphs of order q from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound q on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin. [ABSTRACT FROM AUTHOR]

In 1989, Rota conjectured that, given n bases B 1 , ... , B n of the vector space F n over some field F , one can always decompose the multi-set B 1 ∪ ⋯ ∪ B n into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases B 1 , ... , B n . More specifically, our first result shows that Rota's basis conjecture holds with probability 1 − o (1) as n → ∞ if the bases B 1 , ... , B n are chosen independently uniformly at random among all bases of F q n for some finite field F q (the analogous result is trivially true for an infinite field F). In other words, the conjecture is true for almost all choices of bases B 1 , ... , B n ⊆ F q n . Our second, more general, result concerns random bases B 1 , ... , B n ⊆ S n for some given finite subset S ⊆ F (in other words, bases B 1 , ... , B n where all vectors have entries in S). We show that when choosing bases B 1 , ... , B n ⊆ S n independently uniformly at random among all bases that are subsets of S n , then again Rota's basis conjecture holds with probability 1 − o (1) as n → ∞. [ABSTRACT FROM AUTHOR]

Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most 3. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of (pseudo-)circle arrangements. In this paper the conjecture is verified for △ - saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which maps △ -saturated arrangements with a pentagonal face to arrangements with 4-chromatic 4-regular arrangement graphs. This corona construction has similarities with the crowning construction introduced by Koester (1985). Based on exhaustive experiments with small arrangements we propose three strengthenings of the original conjecture. We further investigate fractional colorings. It is shown that the arrangement graph of every arrangement A of pairwise intersecting pseudocircles is "close" to being 3-colorable. More precisely, the fractional chromatic number χ f (A) of the arrangement graph is bounded from above by χ f (A) ≤ 3 + O (1 n) , where n is the number of pseudocircles of A. Furthermore, we construct an infinite family of 4-edge-critical 4-regular planar graphs which are fractionally 3-colorable. This disproves a conjecture of Gimbel, Kündgen, Li, and Thomassen (2019). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the three dimensional inviscid Boussinesq system in the periodic setting. For 1 ≤ p ≤ ∞ , we show that the threshold regularity exponent for L p -norm conservation of temperature of this system is 1/3, consistent with Onsager exponent. More precisely, for 1 ≤ p ≤ ∞ , every weak solution (v , θ) ∈ C t C x β to the inviscid Boussinesq equations satisfies that ‖ θ (t) ‖ L p (T 3) = ‖ θ 0 ‖ L p (T 3) if β > 1 3 , while if β < 1 3 , there exist infinitely many weak solutions (v , θ) ∈ C t C x β such that the L p -norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in C t C x β for β < 1 3 displaying wild behavior, such as fast kinetic energy dissipation and high oscillation of velocity. Moreover, we also show that if a weak solution (v , θ) of this system has at least one interval of regularity, then this weak solution (v , θ) is not unique in C t C x β for β < 1 3. [ABSTRACT FROM AUTHOR]

We discuss a categorical approach to the theory of discriminants in the combinatorial language introduced by Gelfand, Kapranov and Zelevinsky. Our point of view is inspired by homological mirror symmetry and provides K -theoretic evidence for a conjecture presented by Paul Aspinwall in a conference talk in Banff in March 2016 and later in a joint paper with Plesser and Wang. [ABSTRACT FROM AUTHOR]

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the (2 k + 1) -th moment to the zeroeth moment as the size of the finite field F p r goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure. [ABSTRACT FROM AUTHOR]

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in P 4 (F q 2 ) has at most d (q 5 + q 2) + q 3 + 1 points in common with a threefold of degree d defined over F q 2 . He proved the conjecture for d = 2. In this paper, we show that the conjecture is true for d = 3 and q ≥ 7. [ABSTRACT FROM AUTHOR]

In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column. [ABSTRACT FROM AUTHOR]

This is a corrigendum to the article: "Computation of maximal projection constants" (J. Funct. Anal., 277). The statement of Lemma 3.1(2) of that paper is incorrect. As a consequence of this the proof of Theorem 1.4 is incomplete. In this corrigendum we prove a corrected version of Lemma 3.1 and explain why, with this weaker result, our original strategy for proving Theorem 1.4 no longer works. The other results of the article, in particular the alternative proof of Grünbaum's conjecture, do not depend on Lemma 3.1 and are thus not affected by this error. [ABSTRACT FROM AUTHOR]

The ghost conjecture is first provided by Bergdall and Pollack in [1,2] to study the U p -slopes of spaces of modular forms, which, so far, has already brought plenty of important results. The local version of this conjecture under genericity condition has been solved by Liu-Truong-Xiao-Zhao in [10,11]. In the current paper, we prove a necessary and sufficient condition for a sequence of local ghost series to satisfy that their product has the same Newton polygon to the ghost series build from the direct sum of their associated modules (see Theorem 1.2). That answers a common question asked in both [2,10]. [ABSTRACT FROM AUTHOR]

In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Cheeger–Chern–Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic K -theory of C. [ABSTRACT FROM AUTHOR]

Brualdi–Hoffman–Turán-type problem asks what is the maximum spectral radius λ 1 (G) of an H -free graph G on m edges? This problem gives a spectral perspective on the existence of a subgraph H. A significant result, due to Nikiforov, states that λ 1 (G) ≤ 2 m (1 − 1 r) for every K r + 1 -free graph G (Nikiforov, 2002). Bollobás and Nikiforov further conjectured λ 1 2 (G) + λ 2 2 (G) ≤ 2 m (1 − 1 r) for every K r + 1 -free graph G (Bollobás and Nikiforov, 2007). Let C k + denote the graph obtained from a k -cycle by adding a chord between two vertices of distance two. Zhai, Lin and Shu conjectured that for k ≥ 2 and m sufficiently large, if G is a C 2 k + 1 -free or C 2 k + 2 -free graph, then λ 1 (G) ≤ k − 1 + 4 m − k 2 + 1 2 , with equality if and only if G ≅ K k ∇ (m k − k − 1 2) K 1 (Zhai et al., 2021). This conjecture was also included in a survey of Liu and Ning as one of twenty unsolved problems in spectral graph theory. Recently, Y.T. Li posed a stronger conjecture, which states that the above spectral bound holds for C 2 k + 1 + -free and C 2 k + 2 + -free graphs. In this paper, we confirm these two conjectures by using k -core method and spectral techniques. This presents a new approach to study Brualdi–Hoffman–Turán problems [ABSTRACT FROM AUTHOR]

A well known conjecture of Burr and Erdős asserts that the Ramsey number r (Q n) of the hypercube Q n on 2 n vertices is of the order O (2 n). In this paper, we show that r (Q n) = O (2 2 n − c n) for a universal constant c > 0 , improving upon the previous best known bound r (Q n) = O (2 2 n) , due to Conlon, Fox and Sudakov. [ABSTRACT FROM AUTHOR]

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Their work has opened up a new study of truncated series. Recently, Ballantine and Merca posed a conjecture on infinite families of inequalities involving b 6 (n) , which counts the number of 6-regular partitions of n. In this paper, we confirm Ballantine and Merca's conjecture on linear equalities of b 6 (n) based on a formula of the number of partitions of n into parts not exceeding 3 due to Cayley. [ABSTRACT FROM AUTHOR]

In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function M k (n) to count the number of partitions of n in which k is the least integer that is not a part and there are more parts exceeding k than there are below k. In recent years, two conjectures concerning M k (n) on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large n whenever k is fixed. [ABSTRACT FROM AUTHOR]

By the reduction theorems of Navarro–Tiep and Späth, a way to prove the Alperin weight conjecture and its blockwise version is to verify the so-called inductive Alperin weight condition and inductive blockwise Alperin weight condition for all finite simple groups respectively. In this paper, we establish the inductive blockwise Alperin weight condition for simple groups of type B and odd primes, using a criterion given by Brough and Späth recently. [ABSTRACT FROM AUTHOR]

In this paper, we study higher dimensional anabelian geometry. We prove the Grothendieck conjecture for certain types of hyperbolic polycurves (i.e., algebraic varieties in the form of successive fibrations by hyperbolic curves) over a sub- p -adic field (i.e., a field isomorphic to a subfield of a field finitely generated over the p -adic field Q p) for some prime number p. Among other things, we prove the Grothendieck conjecture for hyperbolic polycurves obtained as certain types of finite étale covers of the moduli space of curves of genus two with ordered r marked points over a sub- p -adic field. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup Q of G is defined, by Glauberman, to be a centrally large subgroup , or CL- subgroup , of G if | Q | ⋅ | Z (Q) | ⩾ | Q ⁎ | ⋅ | Z (Q ⁎) | for every subgroup Q ⁎ of G. A CL-subgroup of G that is minimal under inclusion is called a minimal CL -subgroup of G. Glauberman (2006) [10] studied some properties of CL-subgroups, especially minimal CL-subgroups. He showed that if Q 1 and Q 2 are a pair of minimal CL-subgroups, then (⁎) Q i = (Q 1 ∩ Q 2) Z (Q i) for i = 1 , 2. We say that two CL-subgroups Q 1 and Q 2 (not necessarily minimal) are twisted CL -subgroups if they satisfy the above property (⁎). In this paper, we extend Glauberman's work to prove some further properties of CL-subgroups, especially twisted CL-subgroups. In addition, we give a negative answer to a conjecture proposed by Scoppola et al. (2018) [18]. [ABSTRACT FROM AUTHOR]

In this paper we consider a tower of number fields ⋯ ⊇ K (1) ⊇ K (0) ⊇ K arising naturally from a continuous p -adic representation of Gal (Q ¯ / K) , referred to as a p -adic Lie tower over K. A recent conjecture of Daqing Wan hypothesizes, for certain p -adic Lie towers of curves over F p , a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the p -adic valuation of the discriminant of the extension K (i) / K is given by a polynomial in i , p i for i sufficiently large. This generalizes a previously known result on discriminant-growth in Z p -towers of local fields of characteristic zero. [ABSTRACT FROM AUTHOR]

Abstract Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ (G , X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of n α λ n for some integer α ≥ 0 and some λ ≥ 1. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group F , its d. c. is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are "uniformly small". [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg (2010) [11] have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of G / N G (T) for a split reductive group scheme G and the normalizer of a split maximal torus N G (T) over a field is 1 in the Grothendieck-Witt ring with the characteristic exponent of the field inverted, under the assumption that the base field contains a − 1. Numerous applications of this to splittings in the motivic stable homotopy category and to Algebraic K-Theory are worked out in several related papers by Gunnar Carlsson and the authors. [ABSTRACT FROM AUTHOR]