16 problems

Let $\overrightarrow{F}$ be an antidirected tree with $k$ arcs, meaning that every vertex of $\overrightarrow{F}$ is either a source or a sink. Write…

For a fixed $d$-complex $S$, let $\mathcal{H}(S)$ be the collection of homeomorphs of $S$, and define $\lambda_d$ as the minimum exponent such that … for every fixed $d$-complex…

Mubayi–Zhao conjecture. For $s\geqslant 5$, one has

For a graph $K_k$, let $\mathcal S^rK_k$ denote its $r$-uniform suspension, obtained by adding the same $r-2$ new vertices to every edge of $K_k$. Write…

The bipartite graph gluing conjecture. One has

Let $F_5$ be the 3-uniform hypergraph considered in the $(2,p)$-norm Turán problem, and let $\rho_{2,p}(F_5)$ denote its asymptotic extremal value. For a Steiner triple system (STS…

Erdős's conjecture. Among all triangle-free graphs on $n$ vertices, the maximum number of copies of $C_5$ is attained by the balanced blowup of $C_5$.

Let $d$ be a positive integer, and let $S^d$ denote the simplicial $d$-sphere. Write $\operatorname{ex}(n,S^d)$ for the maximum number of facets in a $d$-dimensional simplicial com…

Localized hypergraph clique-weight conjecture. One has

Frohmader's localized clique-weight conjecture. For every $m$-edge graph $G$,

Regular Turán bound. For such a graph $F$,

Hypergraph Sidorenko conjecture. The hypergraph $X$ contains at least

Erdős's conjecture. In 1979, Erdős conjectured that for every rational $r$ between $1$ and $2$, there exists a finite family of bipartite graphs $\mathcal{F}$ such that

Let $K_{l+1}^+$ denote the $r$-uniform expansion of the complete graph $K_{l+1}$, and let $T_r(n,l)$ be the complete $l$-partite $r$-graph on $n$ vertices whose part sizes differ b…

Finite Turán threshold conjecture. Every poset has a finite Turán threshold, that is, $\pi^{*}(P)<\infty$ for every finite poset $P$.

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be the Boolean lattice, and let ${\rm La}(n,P)$ denote the maximum size of a family of subsets of $[n]$ containing no extension of the poset…