8 problems

Let $c$ satisfy $1/2<c\leqslant1$, and let $G$ be an $n$-vertex graph with minimum degree $\delta(G)\geqslant cn$. Colourful tree embedding conjecture. There exists a constant…

Let $k$ be a positive integer. For a graph $G$ and a vertex $x\in V(G)$, let $N_G(x)$ be the neighbourhood of $x$, and let $N_G^2(x)$ be the set of vertices other than $x$ sharing…

Exact bipartite Komlós–Sárközy–Szemerédi conjecture. For each $\gamma>0$, there are $c,n_0>0$ such that for every $n\ge n_0$, if $G=(A,B,E)$ is a balanced bipartite graph on $2n$ v…

Let $D$ be a digraph, let $k$ be a positive integer, and let $T$ be an antidirected tree with $k$ arcs. Addario-Berry–Havet–Linhares Sales–Reed–Thomassé conjecture. If $D$ has more…

Let $k\in\mathbb{N}$, and let $G$ be a graph. For a vertex $x\in V(G)$, let $N(x)$ denote its neighbourhood and let $N_2(x)$ denote its second neighbourhood. Large first- and secon…

Let $k\in\mathbb{N}$, let $0\leq\alpha\leq \frac{1}{2}$, and let $G$ be a graph. Write $\delta(G)$ and $\Delta(G)$ for the minimum and maximum degrees of $G$, respectively. Interpo…

Let $k,\Delta\in\mathbb{N}$, and let $G$ be a graph. Write $\delta(G)$ and $\Delta(G)$ for the minimum and maximum degrees of $G$, respectively. Constant-degree-tree conjecture. If…

Let $G$ be a graph on $n$ vertices, and let $k$ be a positive integer. A tree with $k$ edges has $k+1$ vertices. Komlós–Sós conjecture. If at least $n/2$ vertices of $G$ have degre…