8 problems

Lovász–Simonovits clique density conjecture. For every $r\ge3$ and every graph $G$,

Lazebnik's conjecture. For all $n \geq r \geq 2$ and $q \geq r$, the Turán graph $T_r(n)$ is the only graph on $n$ vertices and $t_r(n)$ edges that attains the maximum number of…

For $n\ge t$, let $T_{n,t}$ be the $n$-vertex complete $t$-partite Turán graph whose part sizes differ by at most one. For a graph $G$, a $t$-clique cover is a collection of clique…

Let $K_r(t):=T_r(rt)$ be the complete $r$-partite graph with $r$ parts of size $t$, and let $f(n,K_r(t))=I(K_r(t),n)$ denote the maximum induced density of $K_r(t)$ in an $n$-verte…

Let $G$ be an $n$-vertex graph, and define its bisection deficit by … where $\operatorname{bw}(G)$ is the minimum number of edges crossing a balanced partition of the vertex set. S…

Far-from-Turán eigenvalue conjecture. For every $\varepsilon>0$, there exists $c>0$ such that, if $G$ is a regular $n$-vertex graph that is $\varepsilon$-far from every Turán graph…

Let $g_3(n,e)$ be the minimum number of triangles in an $(n,e)$-graph. Let $\mathcal{H}_0^*(n,e)$ and $\mathcal{H}_2^*(n,e)$ denote the subclasses of $\mathcal{H}_0(n,e)$ and…

Let $g_3(n,e)$ denote the minimum number of triangles in an $n$-vertex graph with $e$ edges, and let $t_2(n)$ be the number of edges in the Turán graph $T_2(n)$. Erdős's conjecture…