Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

Sharp thresholds for certain Ramsey properties of random graphs

Sharp thresholds for hypergraph regressive Ramsey numbers

The f -regressive Ramsey number R f (d, n) is the minimum N such that every colouring of the d-tuples of an N -element set mapping each x1, . . . , xd to a colour ≤ f(x1) contains a min-homogeneous set of size n, where a set is called min-homogeneous if every two d-tuples from this set that have the same smallest element get the same colour. If f is the identity, then we are dealing with the st...

Nearly bipartite graphs

We prove that if a nonbipartite graph G on n vertices has minimal degree δ ≥ n 4k + 2 + ck,m where ck,m does not depend on n and n is sufficiently large, if C2s+1 ⊂ G for some k ≤ s ≤ 4k + 1 then C2s+2j+1 ⊂ G for every j = 1, ...,m. We give a structural description of all graphs on n vertices with

Asymptotic Size Ramsey Results for Bipartite Graphs

We show that limn→∞ r̂(F1,n, . . . , Fq,n, Fp+1, . . . , Fr)/n exists, where the bipartite graphs Fq+1, . . . , Fr do not depend on n while, for 1 ≤ i ≤ q, Fi,n is obtained from some bipartite graph Fi with parts V1 ∪ V2 = V (Fi) by duplicating each vertex v ∈ V2 (cv + o(1))n times for some real cv > 0. In fact, the limit is the minimum of a certain mixed integer program. Using the Farkas Lemma ...

Some Ramsey numbers for complete bipartite graphs

Upper bounds are determined for the Ramsey number ](m,n), 2 :; m n. These bounds are attained for infinitely many n in case of m 3 and are fairly close to the exact value for every m if n is sufficiently large.