TY - JOUR

T1 - Ramsey–Turán problems with small independence numbers

AU - Balogh, József

AU - Chen, Ce

AU - McCourt, Grace

AU - Murley, Cassie

N1 - Publisher Copyright:
© 2023

PY - 2024/5

Y1 - 2024/5

N2 - Given a graph H and a function f(n), the Ramsey–Turán number RT(n,H,f(n)) is the maximum number of edges in an n-vertex H-free graph with independence number at most f(n). For H being a small clique, many results about RT(n,H,f(n)) are known and we focus our attention on H=Ks for s≤13. By applying Szemerédi's Regularity Lemma, the dependent random choice method and some weighted Turán-type results, we prove that these cliques have the so-called phase transitions when f(n) is around the inverse function of the off-diagonal Ramsey number of Kr versus a large clique Kn for some r≤s.

AB - Given a graph H and a function f(n), the Ramsey–Turán number RT(n,H,f(n)) is the maximum number of edges in an n-vertex H-free graph with independence number at most f(n). For H being a small clique, many results about RT(n,H,f(n)) are known and we focus our attention on H=Ks for s≤13. By applying Szemerédi's Regularity Lemma, the dependent random choice method and some weighted Turán-type results, we prove that these cliques have the so-called phase transitions when f(n) is around the inverse function of the off-diagonal Ramsey number of Kr versus a large clique Kn for some r≤s.

UR - http://www.scopus.com/inward/record.url?scp=85177747812&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85177747812&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2023.103872

DO - 10.1016/j.ejc.2023.103872

M3 - Article

AN - SCOPUS:85177747812

SN - 0195-6698

VL - 118

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103872

ER -