6 problems

Generalized $k$-independence conjecture. For every $n$-vertex simple graph $G$,

Let $s\geq 2$ and let $\Gamma(n,1,n-s)$ be the flag graph considered in the paper, with vertices the relevant flags of finite sets and adjacency as defined there. The notation $n$…

Let $G=C_5$ be the cycle graph on five vertices, let $n$ be a positive integer, and let $G^{\boxtimes n}$ denote the $n$-fold strong product of $G$. Sonnemann's conjecture. … This…

Spencer's conjecture. The conclusion of the Ajtai–Komlós–Pintz–Spencer–Szemerédi theorem should hold for linear hypergraphs, namely

Let $J_{\pm}(n,4,1)$ be the Johnson-type graph with parameters $n$, $4$, and $1$, and let $\alpha[G]$ denote the independence number of a graph $G$. Independence number conjecture.…

Ramsey–Turán exponent conjecture. The factor $\varepsilon$ can be removed: for every $1/2<\delta<1$,