On an Anti - Ramsey Problem of Burr , Erdős , Graham , and T . Sós Gábor

On an anti-Ramsey problem of Burr, Erdös, Graham, and T. Sós

Anti-ramsey Numbers for Disjoint Copies of Graphs

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph G and a positive integer n, the anti-Ramsey number ar(n, G) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H. Anti-Ramsey numbers were introduced by Erdős, Simonovits and Sós and studied in numerous papers. Let G be a graph with anti-Ramsey number ar(n,...

A small step forwards on the Erdős-Sós problem concerning the Ramsey numbers R(3, k)

Let ∆s = R(K3,Ks) − R(K3,Ks−1), where R(G,H) is the Ramsey number of graphs G and H defined as the smallest n such that any edge coloring of Kn with two colors contains G in the first color or H in the second color. In 1980, Erdős and Sós posed some questions about the growth of ∆s. The best known concrete bounds on ∆s are 3 ≤ ∆s ≤ s, and they have not been improved since the stating of the pro...

On Bipartite Graphs with Linear Ramsey Numbers

We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most ∆ is less than 8(8∆)|V (H)|. This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all ∆≥1 and n≥∆+1 there exists a bipartit...

An Extension Of The Ruzsa-Szemerédi Theorem

We let G(n,m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f (n,p,s) is the smallest m such that every member of G(n,m) contains a member of G(p,s). In this paper we are interested in fixed values r,p and s for which f (n,p,s) grows quadratically with n. A probabilistic construction of Brown, Erdős and T. Sós ([2]) implies that f (n,s(r−2)+2, s)=Ω(n). In the other di...