More than forty years ago, Erdős conjectured that for any t ≤ n k , every k-uniform hypergraph on n vertices without t disjoint edges has at most maxedges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all t < n 3k 2 . This improves upon the best previously known range t = O n k 3 , which dates back to the 1970's.

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R > 1 edges at a time, establishing a general upper bound of n/α√ log α n , where α = 1 + 1 R , thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an n-vertex graph can be as large as n 1− 1 R−2 for finite R, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is at most O(n(log log n) 2 / log n). Our approach is based on expansion.

Let PG(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing PG(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing PG(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach that enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q 4 and sufficiently large m < κqn 2 , where κq ≈ 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m n 2 /4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.

Let i t (G) be the number of independent sets of size t in a graph G. Engbers and Galvin asked how large i t (G) could be in graphs with minimum degree at least δ. They further conjectured that when n ≥ 2δ and t ≥ 3, i t (G) is maximized by the complete bipartite graph K δ,n−δ . This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with m edges and minimum outdegree at least two admits a bipartition in which at least (16+o(1))m edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to 15. If the minimum outdegree tends to infinity with n, then the constant increases to 14. All of these constants are best‐possible, and provide asymptotic answers to a question of Alex Scott. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 147–170, 2016