Two conjectures in Ramsey-Tur\'an theory

TL;DR

This paper advances Ramsey-Turán theory by confirming conjectures for specific graphs and independence numbers, revealing phase transitions, and employing probabilistic and packing techniques to determine maximum edge counts.

Contribution

It proves new exact values and bounds for Ramsey-Turán functions for certain graphs, confirming longstanding conjectures and identifying phase transitions.

Findings

Determined RT(n,K_3,K_s,δn) for s=3,4,5 with small δ

Established RT(n,K_8,o(√n log n)) = n^2/4 + o(n^2)

Identified phase transition at f(n)=Θ(√n log n) for RT(n,K_8)

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 $\phi:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Tur\'an 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 $\textrm{RT}(n,K_3,K_s,\delta n)$ for $s\in\{3,4,5\}$ and sufficiently small $\delta$, confirming a conjecture of Erd\H{o}s and S\'os from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=\Theta(\sqrt{n\log n})$. 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.