13 problems

Polynomial graph-partition conjecture. If $\mathcal{H}$ has bounded VC$_2$ dimension, then it has an $\varepsilon$-graph partition with $M=\operatorname{poly}(1/\varepsilon)$ many…

Let $\mathcal{H}$ be a $3$-graph with bounded VC$_2$ dimension, and let $\mathcal{Q}_V$ denote the vertex partition in the upper-bound theorem. Double-tower upper-bound conjecture.…

Let $(\eta,\psi)$-quasirandom cylinder partitions be partitions of a tripartite $3$-graph satisfying the quasirandomness condition used in the cylinder regularity lemma, and let…

Equitable regularity conjecture. The equitable semialgebraic regularity lemma holds with partitions into

Let $\mathcal{H}$ be a hereditary property of $3$-uniform hypergraphs with infinite $\operatorname{VC}_2$-dimension. Let $L_{\mathcal{H}}(\epsilon_1,\epsilon_2)$ denote the associa…

Let $H$ be a $3$-uniform hypergraph of bounded $\textnormal{VC}_2$-dimension, and consider regular decompositions whose parameter $t$ bounds the number of parts in the underlying p…

Let $G^3(n,p)$ be the random $3$-uniform hypergraph on vertex set $[n]=\{1,\ldots,n\}$, where each triple in $\binom{[n]}{3}$ is included independently with probability $p$. A subh…

For $k,n\geq1$, let $\textnormal{HP}_k(n)$ be the $k$-graph defined by the displayed hyperplane-order construction, and let $\mathcal{H}_{\mathrm{HP}_k}$ be its closure under isomo…

Let $\mathcal{L}={R(x,y,z)}$ be the language of ternary $3$-graph relations, and let $\text{XOP}_2$ be a proposed definition given by an infinite scheme of existential sentences. X…

Let $\mathcal{H}_{\mathrm{HP}}$ and $\mathcal{H}_{\mathrm{GS}_p}$ denote the hereditary $3$-graph properties constructed from the hyperplane-order and Green–Sanders examples. Under…

Let $\mathcal{H}$ be a hereditary $3$-graph property. It is weakly stable if there is some $k$ such that every $3$-graph in $\mathcal{H}$ is weakly $k$-stable, where weakly $k$-sta…

Let $\mathcal{H}$ be a hereditary $3$-graph property. For $k\geq 1$, let $\overline{H}(k)$ be the $3$-graph with vertex set ${a_i,b_i,c_i:i\in[k]}$ and edge set…

Let a hereditary $3$-graph property be a class of finite $3$-uniform hypergraphs closed under isomorphism and induced subgraphs. A decomposition is the type of decomposition appear…