7 problems

KNRS conjecture. Let $H$ be a graph. For every $d,\eta\in(0,1)$, there exists $\varepsilon=\varepsilon(d,\eta,H)$ such that if $G$ is $(\varepsilon,d)$-dense, then

Let $G$ be a graph with vertex set $V(G)$, maximum degree $\Delta$, chromatic index $\chi'(G)$, and fractional chromatic index $\chi'_f(G)$. Hilton's Overfull Conjecture. If … then…

Locally dense graph homomorphism conjecture. For every graph $G$ and every $\delta,d\in(0,1)$, there exists $\varepsilon=\varepsilon(\delta,d,G)$ such that every sufficiently large…

Let $G$ be a graph on $n$ vertices with edge density $\varepsilon$, meaning that $G$ has $\varepsilon\binom{n}{2}$ edges, up to the interpretation of the source's phrase “…

Sárközy--Selkow--Szemerédi conjecture. In the lower bound $h(G)\geq c^n n!$ for Dirac graphs, the constant $c$ can be improved to $1/2-o(1)$; equivalently, every Dirac graph has at…

Kühn--Lapinskas--Osthus conjecture. $G$ contains at least

Let $G$ be a graph, let $|G|$ denote its number of vertices, and let a $K_5$-model be a model of the complete graph $K_5$ in $G$. For a real number $\varepsilon>0$, suppose that th…