17 problems

Mubayi–Rödl conjecture.

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 $r=3$,…

For each integer $r\geq 4$, let $C^{(r)}_{r\ell}$ denote the $r$-uniform tight cycle with $r\ell$ edges, and let $s(F)$ denote the Sidorenko exponent of a hypergraph $F$. Conjectur…

Let $C^{(3)}_{3\ell}$ denote the $3$-uniform tight cycle with $3\ell$ edges, and let $s(F)$ denote the Sidorenko exponent of a hypergraph $F$. Near-Sidorenko conjecture for 3-unifo…

Small-length Turán density conjecture. Both equalities should already hold with $\ell=5$; equivalently, the asserted formulas are conjectured for every $\ell\geq 5$ not divisible b…

The extremal construction conjecture. Under these conditions,

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, with its edges coloured using $r$ colours. A collection of cycles covers $(1-o(1))n$ vertices if the number of uncovered ver…

Let $K_n^{(k)}$ denote the complete $k$-uniform hypergraph, and let its edges be coloured with $r$ colours. A monochromatic tight-cycle partition is a partition of the vertex set i…

Haxell–Łuczak–Peng–Rödl–Ruciński–Skokan conjecture. We have

Let $\mathcal{C}_5^3$ be the $3$-uniform tight cycle on $5$ vertices, and let $\mathcal{C}_5^{-}$ be the $3$-uniform hypergraph obtained from $\mathcal{C}_5^3$ by removing one edge…

Haxell–Łuczak–Peng–Rödl–Ruciński–Skokan conjecture. The Ramsey number satisfies

Let the tight cycle of length $3k$ be the $3$-uniform hypergraph obtained by taking consecutive triples cyclically. Conlon's conjecture. There exists a constant $c>0$ such that the…

Let $r\geq 2$ and let $C^{(r)}_{\ell}$ denote the $r$-uniform tight cycle of length $\ell$. Conlon's conjecture. There exists $c=c(r)>0$ such that, for every $\ell\geq r+1$ divisib…

Let $r\geq 2$, let $\mathcal{C}^{(r)}$ be the family of $r$-uniform tight cycles, and let $S^{(r)}_n$ be the $r$-uniform hypergraph on $[n]$ whose edges are the $r$-element subsets…

Let $(k,s)$ be an admissible pair with $k\ge 3$ odd and $s\ge 5k^2$, and let $t_{k-1}(n,C_s)$ denote the minimum $(k-1)$-degree forcing a perfect $C_s$-tiling in a $k$-uniform hype…

Let $k\ge 3$ and $s\ge k$ be such that $(k,s)$ is an admissible pair, and let $t_{k-1}(n,C_s)$ denote the minimum $(k-1)$-degree forcing a perfect $C_s$-tiling in a $k$-uniform hyp…

Let $C_s^3$ denote the $3$-uniform tight cycle of length $s$, and let $\operatorname{ex}_2(n,C_s^3)$ be the maximum number of edges in a $C_s^3$-free $3$-uniform hypergraph on $n$…