On the Ramsey–Turán number with small s-independence number

Abstract

Let s be an integer, f = f (n) a function, and H a graph. Define the Ramsey-Turán number RT s (n, H, f ) as the maximum number of edges in an H-free graph G of order n with α s (G) < f , where α s (G) is the maximum number of vertices in a K s -free induced subgraph of G. The Ramsey-Turán number attracted a considerable amount of attention and has been mainly studied for f not too much smaller than n. In this paper we consider RT s (n, K t , n δ ) for fixed δ < 1. We show that for an arbitrarily small ε > 0 and 1/2 < δ < 1, RT s (n, K s+1 , n δ ) = Ω(n 1+δ−ε ) for all sufficiently large s. This is nearly optimal, since a trivial upper bound yields RT s (n, K s+1 , n δ ) = O(n 1+δ ). Furthermore, the range of δ is as large as possible. We also consider more general cases and find bounds on RT s (n, K s+r , n δ ) for fixed r ≥ 2. Finally, we discuss a phase transition of RT s (n, K 2s+1 , f ) extending some recent result of Balogh, Hu and Simonovits.

Embedding graphs into larger graphs: results, methods, and problems

“…In other words, sometimes when f (n) is replaced by a slightly smaller g(n), the Ramsey-Turán function noticeably drops. Such result can be found, e.g., in Sudakov [785], in Balogh, Hu, and Simonovits [69], or in Bennett and Dudek [100],. .…”

“…This is possible and often needed, e.g., in the Erdős-Hajnal-Sós-Szemerédi extension of Theorem 5.21 (in [295]), and more generally, this was used in Ramsey type theorems and Ramsey-Turán type theorems, and later in many similar cases. 100 As we have mentioned, this is not quite true. It was invented to prove a conjecture of Bollobás, Erdős, and Simonovits on the parametrized Erdős-Stone theorem, and was first used in the paper of Chvátal and Szemerédi [188].…”

“…"Szemerédi's regularity lemma was first used in his celebrated proof of the Erdős-Turán Conjecture on arithmetic progressions in dense sets of integers. 100 Since then, the lemma has emerged as a fundamental tool in Graph Theory: it has many applications in Extremal Graph Theory, in the area of 'Property Testing' in computer science, combinatorial Number Theory, etc. .…”

Geometric constructions for Ramsey-Turán theory

“…So for q = 4, there is a phase transition somewhere between these functions. See also [9,20] for other results of this type.…”