Two conjectures in Ramsey-Tur\'an theory

Given graphs $H_1,\dots ,H_k$, a graph $G$ is $(H_1,\dots ,H_k)$-free if there is a $k$-edge-colouring $\varphi :E(G)\to [k]$ 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 $\text{RT}(n,H_1,\dots ,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\dots ,H_k)$-free graph with independence number at most $f(n)$. We determine $$\text{RT}$(n,K_3,K_s,\delta n)$for$s\in{3,4,5}$andsufficientlysmall$\delta$, confirming a conjecture of Erd\H{o}s and S'os from 1979. It is known that $\text{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=\Theta(\sqrt{n\log n})$\mathrm{.}However,thevaluesof$$\text{RT}$(n,K_8, o(\sqrt{n\log n}))$ was not known. We determined this value by proving $$\text{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.