7 problems

Let the $K_{4t}$ family denote the family of limited augmented graphs constructed from the complete graph $K_{4t}$, and let $z_L$ be the corresponding limited augmented Zarankiewic…

Let $A,B$ be two multisets of axis-parallel boxes in $\mathbb R^d$, with $|A|=n$ and $|B|=m$, where $m=\operatorname{poly}(n)$. Let $G_{A,B}$ be their intersection graph, and defin…

Let $t\leqslant s\leqslant r$ and let $G$ be an $n\times n\times n$ tripartite graph. Write $\tau(n)$ for the least additive term such that every such graph with minimum degree at…

For positive integers $t$, let $\operatorname{ex}(n,K_{t,t})$ be the maximum number of edges in an $n$-vertex graph containing no copy of $K_{t,t}$. The bipartite Zarankiewicz expo…

Higher-uniformity Zarankiewicz conjecture. For $k>2$, the exponents of $u$ and $D$ in the semialgebraic Zarankiewicz bound can be decreased so that the bound is tight in the sense…

High-girth graph existence conjecture. For every $\ell \in \mathbb{N}$ and $\alpha \geq 1$, there exists an infinite sequence of $(m,n,a,b)$-graphs of girth larger than $4\ell + 2$…

Let $\mathrm{ex}(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph containing no member of a graph family $\mathcal{F}$ as a subgraph. Let $z(n,C_4)$ denote…