Hong Liu, Two conjectures in Ramsey-Turán theory

Two conjectures in Ramsey-Turán theory
Hong Liu
Mathematics Institute, University of Warwick, Warwick, UK
2018/4/10 Tue 5PM
Given graphs H1,…, Hk, a graph G is (H1,…, Hk)-free if there is a k-edge-colouring of G with no Hi in colour-i for all i in {1,2,…,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H1,…,Hk,f(n)) is the maximum size of an n-vertex (H1,…, Hk)-free graph with independence number at most f(n). We determine rt(n,K3,Ks,δn) for s in {3,4,5} and sufficiently small δ, confirming a conjecture of Erdős and Sós from 1979. It is known that rt(n,K8,f(n)) has a phase transition at f(n)=Θ(√(n\log n)). We prove that rt(n,K8,o(√(n\log n)))=n2/4+o(n2), answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings. Joint work with Jaehoon Kim and Younjin Kim.


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