On Ramsey numbers of uniform hypergraphs with given maximum degree

For every ?>0 and every positive integers Δ and r, there exists C=C(?,Δ,r) such that the Ramsey number, R(H,H) of any r-uniform hypergraph H with maximum degree at most Δ is at most C|V(H)|^1+?.     

The VC dimension of k‐uniform random hypergraphs

A set of vertices is shattered in a hypergraph if any of its subsets is obtained as the intersection of an edge with the set. The VC dimension is the size of the largest shattered subset. Under the binomial model of k‐uniform random hypergraphs, the threshold function for the VC dimension to be larger than a given integer is obtained. The same is done for the testing dimension, which is the largest integer d such that all sets of cardinality d are shattered. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Constructing regular self‐complementary uniform hypergraphs

In this article, we examine the possible orders of t‐subset‐regular self‐complementary k‐uniform hypergraphs, which form examples of large sets of two isomorphic t‐designs. We reformulate Khosrovshahi and Tayfeh–Rezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1‐subset‐regular self‐complementary uniform hypergraphs, and prove that these necessary conditions are sufficient for all k, in the case where t = 1. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 439‐454, 2011     

CO‐irredundant Ramsey numbers for graphs

The study of the CO‐irredundant Ramsey numbers t(n₁, ···, n_k) is initiated. It is shown that several values and bounds for these numbers may be obtained from the well‐studied generalized graph Ramsey numbers and the values of t(4, 5), t(4, 6) and t(3, 3, m) are calculated. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 258–268, 2000     

On a question of Sós about 3‐uniform friendship hypergraphs

The well‐known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a “universal friend”). V. Sós defined an analogous friendship property for 3‐uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal friend. We present new 3‐uniform hypergraphs on 8, 16, and 32 vertices that satisfy the friendship property without containing a universal friend. We also prove that if n ≤ 10 and n ≠ 8, then there are no friendship hypergraphs on n vertices without a universal friend. These results were obtained by computer search using integer programming. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 253–261, 2008     

On Ramsey numbers of fans

It is shown that r(F₂,F_n)=4n+1 for n≥2, and r(F_s,F_n)≤4n+2s for n≥s≥2.     

Gallai–Ramsey numbers for cycles

For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of K_n, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K₃ and H is a cycle of a fixed length.     

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

For a finite set V and a positive integer k with , letting be the set of all k‐subsets of V, the pair is called the complete k‐hypergraph on V, while each k‐subset of V is called an edge. A factorization of the complete k‐hypergraph of index , simply a ‐factorization of order n, is a partition of the edges into s disjoint subsets such that each k‐hypergraph , called a factor, is a spanning subhypergraph of . Such a factorization is homogeneous if there exist two transitive subgroups G and M of the symmetric group of degree n such that G induces a transitive action on the set and M lies in the kernel of this action. In this article, we give a classification of homogeneous factorizations of that admit a group acting transitively on the edges of . It is shown that, for and , there exists an edge‐transitive homogeneous ‐factorization of order n if and only if is one of (32, 3, 5), (32, 3, 31), (33, 4, 5), , and , where and q is a prime power with .     

Disjoint perfect matchings in 3‐uniform hypergraphs

For a hypergraph H, let denote the minimum vertex degree in H. Kühn, Osthus, and Treglown proved that, for any sufficiently large integer n with , if H is a 3‐uniform hypergraph with order n and then H has a perfect matching, and this bound on is best possible. In this article, we show that under the same conditions, H contains at least pairwise disjoint perfect matchings, and this bound is sharp.     

Constrained Ramsey numbers for rainbow matching

The Ramsey number R_k(G) of a graph G is the minimum number N, such that any edge coloring of K_N with k colors contains a monochromatic copy of G. The constrained Ramsey number f(G, T) of the graphs G and T is the minimum number N, such that any edge coloring of K_N with any number of colors contains a monochromatic copy of G or a rainbow copy of T. We show that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK₂) = R_{t ? 1}(G) for t≥2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:91‐95, 2011     

Bipartite anti‐Ramsey numbers of cycles

We determine the maximum number of colors in a coloring of the edges of K_m,n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q≤ a, let g(a,q) be the maximum number of edges in a spanning subgraph G of K_a,a such that the minimum number of vertex‐disjoint even paths and pairs of vertices from distinct partite sets needed to cover V(G) is q. We prove that g(a,q) = a² ? aq + max {a, 2q ? 2}. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 9–28, 2004     

List coloring triangle‐free hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with , and . Johansson [Tech. report (1996)] proved that every triangle‐free graph with maximum degree Δ has list chromatic number . Frieze and Mubayi (Electron J Comb 15 (2008), 27) proved that every linear (meaning that every two edges share at most one vertex) triangle‐free triple system with maximum degree Δ has chromatic number . The restriction to linear triple systems was crucial to their proof. We provide a common generalization of both these results for rank 3 hypergraphs (edges have size 2 or 3). Our result removes the linear restriction from 8 , while reducing to the (best possible) result [Johansson, Tech. report (1996)] for graphs. In addition, our result provides a positive answer to a restricted version of a question of Ajtai Erd?s, Komlós, and Szemerédi (combinatorica 1 (1981), 313–317) concerning sparse 3‐uniform hypergraphs. As an application, we prove that if is the collection of 3‐uniform triangles, then the Ramsey number satisfies for some positive constants a and b. The upper bound makes progress towards the recent conjecture of Kostochka, Mubayi, and Verstraëte (J Comb Theory Ser A 120 (2013), 1491–1507) that where C₃ is the linear triangle. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 487–519, 2015     

On graphs with small Ramsey numbers*

Let R(G) denote the minimum integer N such that for every bicoloring of the edges of K_N, at least one of the monochromatic subgraphs contains G as a subgraph. We show that for every positive integer d and each γ,0 < γ < 1, there exists k = k(d,γ) such that for every bipartite graph G = (W,U;E) with the maximum degree of vertices in W at most d and , . This answers a question of Trotter. We give also a weaker bound on the Ramsey numbers of graphs whose set of vertices of degree at least d + 1 is independent. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 198–204, 2001