A Precise Threshold for Quasi-Ramsey Numbers

Abstract

We consider a variation of Ramsey numbers introduced by Erdős and Pach [6], where instead of seeking complete or independent sets we only seek a t-homogeneous set, a vertex subset that induces a subgraph of minimum degree at least t or the complement of such a graph.For any ν > 0 and positive integer k, we show that any graph G or its complement contains as an induced subgraph some graph H on ℓ ≥ k vertices with minimum degree at least 1 2 (ℓ − 1) + ν provided that G has at least k Ω(ν 2 ) vertices. We also show this to be best possible in a sense. This may be viewed as correction to a result claimed in [6].For the above result, we permit H to have order at least k. In the harder problem where we insist that H have exactly k vertices, we do not obtain sharp results, although we show a way to translate results of one form of the problem to the other.

On a Ramsey-type problem of Erdős and Pach

“…also Theorem 4 in [5]). The astute reader may later notice that the second-order term ν √ ℓ − 1 in the minimum degree guarantee of Theorem 2 can be straightforwardly improved to an Ω( (ℓ − 1) ln ln ℓ) term.…”

“…This is a bound on a variable quasi-Ramsey number which is similar to Theorem 3(a) in [5]. The idea of the proof of this auxiliary result is inspired by the sketch argument for Theorem 2 in [1], in spite of the error contained in that sketch (cf.…”

“…The idea of the proof of this auxiliary result is inspired by the sketch argument for Theorem 2 in [1], in spite of the error contained in that sketch (cf. [5]). …”

“…On the other hand, a standard random graph construction yields the following, which certifies that the secondorder term cannot be improved to a ω( (ℓ − 1) ln ln ℓ) term. [5]. (We may not use Theorem 3(b) in [5] directly as stated as it needs ν(ℓ) to be non-decreasing in ℓ.…”

“…[5]. (We may not use Theorem 3(b) in [5] directly as stated as it needs ν(ℓ) to be non-decreasing in ℓ. )…”