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In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries are i.i.d. copies of a random variable $\xi \in \mathbb{Z}_p$. We consider the minimal number of random entries of $X_n$ required for the cokernel of $X_n$ to converge to the Cohen--Lenstra distribution. When $\xi$ is given by the Haar measure, we prove a lower bound of the number of random entries and prove its converse-type result using random regular bipartite multigraphs. When $\xi$ is a general random variable, we determine the minimal number of random entries. Let $M_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ with $k$-step stairs of zeros and the other entries given by independent random $\epsilon$-balanced variables valued in $\mathbb{Z}_p$. We prove that the cokernel of $M_n$ converges to the Cohen--Lenstra distribution under a mild assumption. This extends Wood's universality theorem on random $p$-adic matrices., Comment: 48 pages

The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? We adapt a proof method of Norin, Scott, Seymour and Wood (2019) to show that the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of $H$.

For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $\delta_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = \Omega(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $\delta_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching. Moreover, for every fixed integer $d < k$ and $\gamma > 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $\delta_d(\mathcal H) \geq m_{d}(k,n) + \gamma\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as ``robust'' versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - \Theta (n))$ many perfect matchings, which is best possible up to an $\exp(\Theta(n))$ factor., Comment: Final version, to appear in Combinatorica (26 pages + 2 page appendix); Theorem 1.5 was proved in independent work of Pham, Sah, Sawhney, and Simkin (arxiv:2210.03064)

We prove that for $n \in \mathbb N$ and an absolute constant $C$, if $p \geq C\log^2 n / n$ and $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k\in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i, j\in [n]$, then asymptotically almost surely there is an order-$n$ Latin square in which the entry in the $i$th row and $j$th column lies in $L_{i,j}$. The problem of determining the threshold probability for the existence of an order-$n$ Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides an upper bound which is tight up to a factor of $\log n$ and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a $1$-factorization of a nearly complete regular bipartite graph., Comment: 32 pages, 1 figure. Final version, to appear in Transactions of the AMS

In 1977, Erd\H{o}s asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the largest possible chromatic number of the union $\bigcup_{i=1}^{n} G_i$? The equivalent dual formulation of this question asks for the largest chromatic index of an $n$-vertex hypergraph with maximum degree at most $n$ and maximum codegree at most $t$. For the case $t = 1$, Erd\H{o}s, Faber, and Lov\'{a}sz famously conjectured that the answer is $n$, which was recently proved by the authors for all sufficiently large $n$. In this paper, we answer this question of Erd\H{o}s for $t \geq 2$ in a strong sense, by proving that every $n$-vertex hypergraph with maximum degree at most $(1-o(1))tn$ and maximum codegree at most $t$ has chromatic index at most $tn$ for any $t,n \in \mathbb{N}$. Moreover, equality holds if and only if the hypergraph is a $t$-fold projective plane of order $k$, where $n = k^2 + k + 1$. Thus, for every $t \in \mathbb N$, this bound is best possible for infinitely many integers $n$. This result also holds for the list chromatic index., Comment: Final version, to appear in Proceedings of the London Mathematical Society (26 pages)

This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random `nibble' methods. We also give a detailed sketch of some aspects of the recent proof of the Erd\H{o}s-Faber-Lov\'{a}sz conjecture., Comment: Final version, to appear in the proceedings of the 8th European Congress of Mathematics; 33 pages, 3 figures

The Erd\H{o}s-Faber-Lov\'{a}sz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn., Comment: 47 pages, 3 figures. Final version, to appear in the Annals of Mathematics

Let $H$ be a $k$-uniform $D$-regular simple hypergraph on $N$ vertices. Based on an analysis of the R\"odl nibble, Alon, Kim and Spencer (1997) proved that if $k \ge 3$, then $H$ contains a matching covering all but at most $ND^{-1/(k-1)+o(1)}$ vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all $k > 3$, $H$ contains a matching covering all but at most $ND^{-1/(k-1)-\eta}$ vertices for some $\eta = \Theta(k^{-3}) > 0$, when $N$ and $D$ are sufficiently large. Our approach consists of showing that the R\"odl nibble process not only constructs a large matching but it also produces many well-distributed `augmenting stars' which can then be used to significantly improve the matching constructed by the R\"odl nibble process. Based on this, we also improve the results of Kostochka and R\"odl (1998) and Vu (2000) on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed (2000) on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs)., Comment: 36 pages, 2 figures

We conduct a quantitative analysis on the number of random edges required to be added to a base graph~$H$ to significantly increase natural minor-monotone graph parameters in the resulting graph~$R$. Specifically, we show that if $R$ is obtained from a connected graph $H$ by adding only a few random edges, the tree-width, genus, and Hadwiger number of $R$ become very large, irrespective of the structure of~$H$., Comment: 20 pages

Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A hypergraph $\mathcal{H}$ is {\it Berge-$\mathcal{F}$-free} if it does not contain a subhypergraph which is Berge copy of $\mathcal{F}$. This is a generalization of the usual, graph based Berge hypergraphs, where $\mathcal{F}$ is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph based setting. In particular, we show that for any $r$-uniform hypregraph $\mathcal{F}$, the sum of the sizes of the hyperedges of a (not necessarily uniform) Berge-$\mathcal{F}$-free hypergraph $\mathcal{H}$ on $n$ vertices is $o(n^r)$ when all the hyperedges of $\mathcal{H}$ are large enough. We also give a connection between hypergraph based Berge hypergraphs and generalized hypergraph Tur\'an problems., Comment: 13 pages

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) = \Theta(n^r)$. Several decades ago, Erd\H{o}s and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$. In this paper, we make progress on the conjecture of Erd\H{o}s and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

K\"uhn, Osthus, and Townsend asked whether there exists a constant $C$ such that every strongly $Ct$-connected tournament contains all possible $1$-factors with at most $t$ components. We answer this question in the affirmative. This is best possible up to constant. In addition, we can ensure that each cycle in the $1$-factor contains a prescribed vertex. Indeed, we derive this result from a more general result on partitioning digraphs which are close to semicomplete. More precisely, we prove that there exists a constant $C$ such that for any $k\geq 1$, if a strongly $Ck^4t$-connected digraph $D$ is close to semicomplete, then we can partition $D$ into $t$ strongly $k$-connected subgraphs with prescribed sizes, provided that the prescribed sizes are $\Omega(n)$. This result improves the earlier result of K\"uhn, Osthus, and Townsend. Here, the condition of connectivity being linear in $t$ is best possible, and the condition of prescribed size being $\Omega(n)$ is also best possible.

Mader proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, where the equality holds for the complete bipartite digraph ${DK}_{k,n-k}$. For dense strongly $k$-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly $k$-connected $n$-vertex digraph $D$ contains a strongly $k$-connected spanning subgraph with at most $kn + 800k(k+\overline{\Delta}(D))$ edges, where $\overline{\Delta}(D)$ denotes the maximum degree of the complement of the underlying undirected graph of a digraph $D$. Here, the additional term $800k(k+\overline{\Delta}(D))$ is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly $k$-connected $n$-vertex semicomplete digraph contains a strongly $k$-connected spanning subgraph with at most $kn + 800k^2$ edges, which is essentially optimal since $800k^2$ cannot be reduced to the number less than $k(k-1)/2$. We also prove an analogous result for strongly $k$-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms., Comment: 31 pages

As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph of bounded maximum degree. Secondly, we prove that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$ sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into $496t$ such sets., Comment: 15 pages

We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs $K_t \boxminus K_t$ and $K_t \boxminus S_t$ are graphs obtained from the disjoint union of two cliques of size $t$, and one clique of size $t$ and one independent set of size $t$ respectively, by adding a perfect matching. We prove that : (1) interval graphs are $(K_3\boxminus S_3)$-free chordal graphs; and $(K_t\boxminus S_t)$-free chordal graphs have mim-width at most $t-1$, (2) permutation graphs are $(K_3\boxminus K_3)$-free co-comparability graphs; and $(K_t\boxminus K_t)$-free co-comparability graphs have mim-width at most $t-1$, (3) chordal graphs and co-comparability graphs have unbounded mim-width in general. We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time $n^{\mathcal{O}(t)}$ on $(K_t\boxminus S_t)$-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless $P=NP$. We generalize these ideas to bigger graph classes. We introduce a new width parameter sim-width, of stronger modelling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding $K_t\boxminus K_t$ and $K_t\boxminus S_t$ as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width., Comment: 24 pages, 5 figures; An extended abstract appeared in the proceedings of WALCOM2017

In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs.

An $r$-matching in a graph $G$ is a collection of edges in $G$ such that the distance between any two edges is at least $r$. A $2$-matching is also called an induced matching. In this paper, we estimate the maximum number of $r$-matchings in a tree of fixed order. We also prove that the $n$-vertex path has the maximum number of induced matchings among all $n$-vertex trees.

A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t} \choose {k-t}}$ if $n\geq n_0(k,t)$. Alon, Aydinian, and Huang considered families generalizing intersecting families, and proved the same bound. In this paper, we give a strengthening of their result by considering families generalizing $t$-intersecting families for all $t \geq 1$. In 2004, Talbot generalized Bollob\'{a}s's Two Families Theorem to $t$-intersecting families. In this paper, we proved a slight generalization of Talbot's result by using the probabilistic method.

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property., Comment: 6 pages