5 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.…

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 $\mathcal{L}={R(x,y,z)}$ 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…