The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl

Monochromatic Trees with Respect to Edge Partitions

AbstractIt is shown, that for every infinite cardinal κ there exists a graph F on κ vertices satisfying F → (T)edgesλ for every tree T on κ vertices and all λ satisfying cfκ → (ω)3

Applications of hypergraph coloring to coloring graphs not inducing certain trees

AbstractWe present a simple result on coloring hypergraphs and use it to obtain bounds on the chromatic number of graphs which do not induce certain trees

An optimal algorithm for checking regularity (extended abstract)

We present a deterministic algorithm A that, in 0(m2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemeredi [18]. In the case in which G is not regular enough, our algorithm outputs a vntness to this irregularity. Algorithm A may be used as a subroutine in an algorithm that finds an e-regular partition of a given n-vertex graph Λ in time 0(n2). This time complexity is optimal, up to a constant factor, and improves upon the bound 0(M(n)), proved by Alon, Duke, Lefmann, Rodl, and Yuster [1, 2], where M(n) = 0(n2-376) is the time required to square a 6-1 matrix over the integers. Our approach is elementary, except that it makes use of linear-sized expanders to accomplish a suitable form of deterministic sampling

Finite Induced Graph Ramsey Theory: On Partitions of Subgraphs

AbstractFor given finite (unordered) graphs G and H, we examine the existence of a Ramsey graph F for which the strong Ramsey arrow F → (G)Hr holds. We concentrate on the situation when H is not a complete graph. The set of graphs G for which there exists an F satisfying F → (G)P22 (P2 is a path on three vertices) is found to be the union of the set of chordal comparability graphs together with the set of convex graphs

Dense Graphs without 3-Regular Subgraphs

AbstractIn this paper, we show the existence of graphs with cn log log n edges that contain no 3-regular subgraphs. On the other hand, we show that graphs with ckn log Δ(G) edges contain k-regular subgraphs. We also consider a related problem for graphs with cn2 edges

Geometrical realization of set systems and probabilistic communication complexity

Let d = d(n) be the minimum d such that for every sequence of n subsets F I, F 2,.•., F n of {I, 2,..., n} there exist n points PI ' P 2,..., P n and n hyperplanes HI ' H 2,..., Hn in R d such that P j lies in the positive side of Hi iff j E Fi. Then In this paper we prove: Theorem 1 1. If n,m-+-00 and log2m = o(n) then d(n,m)s(t + o(l))n Put d = d(n, 1n) the