Posts Tagged ‘HongLiu’

Hong Liu, Two conjectures in Ramsey-Turán theory

Thursday, March 8th, 2018
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.

Hong Liu, On the maximum number of integer colourings with forbidden monochromatic sums

Sunday, December 3rd, 2017
On the maximum number of integer colourings with forbidden monochromatic sums
Hong Liu
Mathematics Institute, University of Warwick, Warwick, UK
2017/12/27 Wed 4PM-5PM (Room 3433)
Let f(n,r) denote the maximum number of colourings of A⊆{1,…,n} with r colours such that each colour class is sum-free. Here, a sum is a subset {x,y,z} such that x+y=z. We show that f(n,2) = 2⌈n/2⌉, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of f(n,r) for r≤5.
Joint work with Maryam Sharifzadeh and Katherine Staden.