32 problems

Let $K_4^{(3)-}$ denote the $3$-graph obtained from the complete $3$-graph on four vertices by deleting one edge, and let $K_4^{(3)}$ denote the complete $3$-graph on four vertices…

Let integers $\ell\geq r\geq 3$, and let ${\mathcal T}_r(n,\ell)$ denote the complete balanced $\ell$-partite $r$-uniform hypergraph. Write $K_{\ell+1}^r$ for the complete $r$-unif…

Let $t\ge4$ be fixed. For each integer $r\ge2$, let $\mathcal D_{r,t}$ be the $(r,t)$-daisy, obtained by adjoining a fixed set of $r-2$ new vertices to every edge of the complete g…

Palette conjecture. Every lower bound for a uniform Turán density should be obtained by a palette construction; equivalently, every such lower bound should arise from the palette L…

Let $f_r(n,v,e)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph such that every set of $v$ vertices spans fewer than $e$ edges. For fixed integers…

Let $P_\ell^+$ denote the expansion of a path $P_\ell$, and let $\mathcal{F}^*_{n,k,\ell}$ be the explicitly defined $P_\ell^+$-free family obtained from the star construction, wit…

Tree-suspension optimality conjecture. Then

Erdős's gap conjecture. Every point of $[0,1)$ has a non-trivial gap immediately to its right in $\Pi^k_{\infty}$.

Let $F$ be a $k$-graph. For integers $1\leq\ell\leq k-3$, let $\pi_{\rm u}^{(\ell)}(F)$ denote its $\ell$-uniform Turán density. An $(\ell+1)$-vanishing order is an ordering of…

Let $F$ be a $(k-1)$-graph, and let $\mathcal{S}_F$ be the $k$-graph obtained by adjoining an apex vertex $v$ and including $v\cup e$ as an edge for every $e\in E(F)$. For…

For integers $k\geq4$ and $\ell\geq3$ with $\ell<k$, let $F$ be a $k$-graph. An $\ell$-vanishing order of $F$ is an ordering of $V(F)$ admitting an $\ell$-vanishing coloring with t…

Let $\mathcal{C}_\ell^r$ be the tight $r$-uniform cycle of length $\ell>r$, with vertex set $\{0,1,\ldots,\ell-1\}$ and edges … where addition is modulo $\ell$. For $r=3$,…

Let $\mathcal{H}$ be an $r$-uniform hypergraph, and for a family $\mathcal{F}$ of $r$-graphs let $\mathrm{ex}_{r-1}(n,\mathcal{F})$ be the maximum possible minimum codegree…

Given an integer $r\geq 3$, let $K_3^r$ be the $r$-uniform expansion of the complete graph $K_3$, and let ${\rm ex}_r(n,K_3^r)$ denote the maximum number of edges in an $r$-uniform…

Let $F_5$ be the 3-uniform hypergraph defined in the paper, and let $\mathcal{H}_1,\ldots,\mathcal{H}_p\subset\binom{[n]}{3}$ be 3-uniform hypergraphs. They are 2-colored $F_5$-fre…

Let $F_5$ denote the 3-uniform hypergraph in the paper, let $M_{s+1}^3$ denote a matching of size $s+1$, and let ${\rm ex}_3(n,\{F_5,M_{s+1}^3\})$ be the maximum number of edges in…

Higher-uniformity extremal construction conjecture. For all sufficiently long $L$ relatively prime to $r$, the extremal $C_L^{(r)}$-hom-free constructions on sufficiently large num…

Natural analogue conjecture. For residues $k$ relatively prime to $r$, the extremal ${\mathscr C}_{k}^{(r)}$-hom-free constructions are the natural analogues of the constructions i…

A $3$-graph is linear if any two distinct edges have at most one vertex in common. A $3$-graph $F$ is layered if there is a function $f:V(F)\to\mathbb N$ satisfying the source's la…

A $3$-graph $F$ is layered if there is a function $f:V(F)\to\mathbb N$ such that: every edge has exactly one vertex whose label is strictly greater than the other two; edges with t…

Let $G^3_{n,p}$ be the random $3$-uniform hypergraph on $n$ vertices in which each $3$-edge is present independently with probability $p$, and let $C^3_4$ be the $3$-uniform linear…

Optimal construction conjecture.

Partially directed triangle conjecture. For all sufficiently large $n$, every $n$-vertex partially directed $k$-graph with $\alpha\binom{n}{k}$ undirected edges and…

Let $F$ be an $r$-uniform hypergraph. When the limit exists, define its relative Turán exponent by … where the supremum is over all $r$-graphs $H$ with maximum degree at most…

Let $r\geq 2$ and let $2\leq s_1\leq s_2\leq\cdots\leq s_r$. Write $K_{s_1,\ldots,s_r}$ for the complete $r$-partite $r$-uniform hypergraph with parts of these sizes, and let…