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arXiv · 1803.04721

Two conjectures in Ramsey-Turán theory

Jaehoon Kim·Younjin Kim·Hong Liu
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Abstract

Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $ϕ:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Turán function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,δn)$ for $s\in\{3,4,5\}$ and sufficiently small $δ$, confirming a conjecture of Erdős and Sós from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=Θ(\sqrt{n\log n})$. However, the values of $\textrm{RT}(n,K_8, o(\sqrt{n\log n}))$ was not known. We determined this value by proving $\textrm{RT}(n,K_8,o(\sqrt{n\log n}))=\frac{n^2}{4}+o(n^2)$, answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings.

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Jaehoon Kim, Younjin Kim, Hong Liu. 2018-03-13. Two conjectures in Ramsey-Turán theory. https://arxiv.org/abs/1803.04721

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