TY - JOUR
T1 - Ramsey–Turán problems with small independence numbers
AU - Balogh, József
AU - Chen, Ce
AU - McCourt, Grace
AU - Murley, Cassie
N1 - Research is partially supported by National Science Foundation Grant DMS-1764123, NSF RTG grant DMS 1937241, Arnold O. Beckman Research Award (UIUC Campus Research Board RB 22000), and the Langan Scholar Fund (UIUC). Grace McCourt was partially funded by RTG grant DMS 1937241, while working on this project.
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.
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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 -