11 problems

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

For $k\geq 3$, let $H_1$ and $H_2$ be $k$-graphs on the same $n$-vertex set, and let $T_k$ denote the generalised triangle. Write $\delta(H_i)$ for the minimum codegree of $H_i$. T…

Let $r\ge 3$ and $1 \le s_1 \le \cdots \le s_r$ be integers, and let $m\coloneqq s_1+\cdots+s_r$. A spanning subgraph $F$ of $K_{s_1,\ldots,s_r}^{r}$ is rigid if, for some such int…

Let $r\ge 2$ and $1 \le s_1 \le \cdots \le s_r$ be integers, and let $m \coloneqq s_1 + \cdots + s_r$. Suppose that $F$ is a spanning subgraph of $K_{s_1, \ldots, s_r}^{r}$. For in…

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…

Lopsided fractional Erdős matching conjecture. For every $k$-graph $F$ and $0\leq\beta\leq 1/v(F)$,

Gan–Han–Sun–Wang conjecture. The displayed asymptotic upper bound should hold. This conjecture generalizes the matching problem to tilings by two-edge hypergraphs with prescribed i…

For $k\ge3$, $1\le\ell<k/2$, and $k-\ell\le d\le k-1$, let $h_d^\ell(k)$ be the limiting normalized minimum $d$-degree threshold for Hamilton $\ell$-cycles. Let $t(k,d,\ell)$ be th…

Let $Y_{k,b}$ be the $k$-graph consisting of two edges intersecting in exactly $b$ vertices. Let $n,s,k,b$ be positive integers such that $k>b>0$ and $n\ge(2k-b)(s+1)-1$. If $H$ is…

$K_4^3$-free matching conjecture. For any $\gamma>0$, for sufficiently large $n\in 3\mathbb{N}$, if $H$ contains no copy of $K_4^3$ and

Let $t_2^3(n,K_4^3-e)$ be the smallest integer $d$ such that every 3-uniform hypergraph on $n$ vertices with minimum 2-degree at least $d$ contains a spanning collection of vertex-…