12 problems

Let $k\geqslant 2$, let $\Delta$ be sufficiently large, and let $H$ be a $k$-partite $k$-graph of maximum degree at most $\Delta$. Write $\chi_\ell(H)$ for the list chromatic numbe…

Simultaneous multipartite subhypergraph conjecture. There exists a partition of $V$ into $r$ classes $V_1,\dots,V_r$ such that, for every $i=1,\dots,\ell$, at least

Dhawan's conjecture. For every $k\geqslant 2$, there is a constant $c=c(k)>0$ such that, for all sufficiently large $\Delta$,

Let $n,s,k$ be positive integers, and let $\mathcal F\subseteq[n]^k$. Two tuples in $[n]^k$ are disjoint if they differ in every coordinate. Partite Erdős matching conjecture. If ……

Let $n,m,k$ be positive integers with $n$ sufficiently large, $n>m$, and $k\geq 4$. Let $H_k(n,m)$ be the $k$-partite $k$-graph constructed by taking pairwise disjoint partition cl…

Linear-growth conjecture. For every $k\geq 3$,

Let $\mathscr K_{m_1,\ldots,m_n}^{3}$ be the complete multipartite 3-uniform hypergraph whose parts have sizes $m_1,\ldots,m_n$, and let $\lambda \mathscr K_{m_1,\ldots,m_n}^{3}$ d…

Let $r\ge 2$, and let $p_1,p_2,\ldots,p_k$ be positive integers. For each $i$, let $V_i$ be a part of size $p_i$, and let $K_{p_1,p_2,\ldots,p_k}^r$ be the $r$-complete $k$-partite…

Equality conjecture. Equality holds in Theorem for all $r$ and $\theta$; equivalently,

Let $H$ be a $k$-partite $k$-graph with parts of size $n$ satisfying the hypotheses of Theorem, and let a certificate be the obstruction defined there. The multipartite certificate…

Let $H$ be an $n$-balanced $r$-partite $r$-graph, and let $I$ be a subset of $[r]:=\{1,2,\ldots,r\}$. An $I$-tuple is an element of $\times_{i\in I}V_i$, and an $I^c$-tuple is defi…

Let $H$) be an $n$-balanced $r$-partite $r$-graph, and let $I$ be a subset of $[r]:=\{1,2,\ldots,r\}$. An $I$-tuple is an element of $\times_{i\in I}V_i$, and write…