19 problems

Let $G$ be a graph, with maximum degree $\operatorname{\Delta}(G)$, maximum codegree $\operatorname{\Delta_2}(G)$, and chromatic number $\chi(G)$. Vu's sharpened conjecture. There…

Mycroft's constant-error conjecture. There exists a constant $C=C(F)$ such that the error terms in $$ can be replaced by $C$.

Let $\mathcal{H}$ be an $r$-uniform hypergraph, and for a family $\mathcal{F}$ of $r$-graphs let $\mathrm{ex}_{r-1}(n,\mathcal{F})$ be the maximum possible minimum codegree…

Let $G$ be a $3$-graph on $n$ vertices, and let $\delta_2(G)$ denote its minimum codegree. Define the $4$-graph $H$ on $V(G)$ by making a $4$-set an edge whenever it spans a tetrah…

Let $G$ be a graph. Its square $G^2$ is obtained by joining distinct vertices that are connected by a two-edge path in $G$. Let $\operatorname{\Delta}(G)$ be the maximum degree,…

Let $G$ be a graph. Write $\operatorname{\Delta_2}(G)$ for its maximum codegree and $\operatorname{\Delta}(G)$ for its maximum degree. Vu's conjecture. Fix…

There exists an integer $n_0$ such that for all $n\geq n_0$, the following holds. Let $\mathcal{C}$ be a $3$-graph consisting of vertex-disjoint loose cycles…

Supported-codegree conjecture for tight Hamilton cycles. For every $k\geqslant 2$, there is some $n_0$ such that every such $G$ with $n\geqslant n_0$ and…

Georgakopoulos et al.'s minimum-codegree conjecture. Every $n$-vertex $k$-graph with minimum codegree at least $n/k$ contains a spanning $(k-1)$-sphere.

Higher-dimensional spanning-sphere conjecture. If $\mathcal{H}$ comprises a single tight component and every set $Z$ of $r-1$ vertices contained in an edge of $\mathcal{H}$ is cont…

Spanning-sphere conjecture. Every such $\mathcal{H}$ with

Rödl–Ruciński–Szemerédi conjecture. If

Covector codegree decomposition conjecture. Every irreducible vector configuration $\boldsymbol{\mathbf{V}}$ of rank $r$ admits a codegree${}^{\star}$ decomposition of length at le…

Lattice join projection conjecture. If

Codegree decomposition conjecture. If

Even-dimensional rational codegree conjecture. In even dimension, the condition

Dickenstein–Nill rational codegree conjecture. If

Let $P$ be a smooth lattice polytope of dimension $n$, and let $\operatorname{codeg}(P)$ denote its codegree. A polytope is ${\mathbb Q}$-normal when its nef value agrees with its…

Let $P$ be a smooth lattice polytope of dimension $n$. Write $\operatorname{codeg}(P)$ for its codegree, $\operatorname{codeg}_{\mathbb Q}(P)$ for its rational codegree, and let…