Hong Liu, Two conjectures in Ramsey-Turán theory

Given graphs H₁,…, H_k, a graph G is (H₁,…, H_k)-free if there is a k-edge-colouring of G with no H_i in colour-i for all i in {1,2,…,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H₁,…,H_k,f(n)) is the maximum size of an n-vertex (H₁,…, H_k)-free graph with independence number at most f(n). We determine rt(n,K₃,K_s,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.