7 problems

Let $\mathcal T_m$ denote the set of labeled trees on $[m]=\{1,\ldots,m\}$. For $T\in\mathcal T_n$ and $U\in\mathcal T_m$, write $T\preceq U$ if $U$ displays $T$, meaning that $T$…

Let $G$ be an $n$-vertex graph, and let $k$ be a positive integer. Sparse degree conjecture. If $\delta(G)\ge\frac k2$ and at least $\frac{n}{2\sqrt{k}}$ vertices of $G$ have degre…

Let $k$ be a positive integer and let $\alpha\in[0,\frac13)$. Intermediate-range conjecture. Every graph with minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at lea…

Let $G$ be a graph, and let $k$ be a positive integer. $2k$–$k/2$ conjecture. If $\delta(G)\ge\frac{k}{2}$ and $\Delta(G)\ge2k$, then $G$ contains every tree with $k$ edges. This i…

Let $G$ be a graph, and let $k$ be a positive integer. Two-thirds conjecture. If $\delta(G)\ge\left\lfloor\frac{2k}{3}\right\rfloor$ and $\Delta(G)\ge k$, then $G$ contains every t…

Let a $k$-edge tree be $r$-skew if one of its colour classes has size at most $r(k+1)$. Skew Loebl–Komlós–Sós conjecture. If a graph $G$ on $n$ vertices has more than $rn$ vertices…

Let $G$ be an $n$-vertex graph, and let $k$ be a positive integer. Sharp Loebl–Komlós–Sós conjecture. If more than…