10 problems

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…

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…

Approximate Lehel conjecture. There exist constants $c_k\geq1$ and $n_0(k)\in\mathbb N$ such that, for every $n\geq n_0(k)$, the coloured complete $k$-graph contains vertex-disjoin…

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…

Let $\mathcal{C}_5^3$ be the $3$-uniform tight cycle of length $5$, with vertex set $\{1,\dots,5\}$ and edges of the form $\{x,x+1,x+2\pmod 5\}$. Let $\pi(\mathcal{C}_5^3)$ denote…

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 $(k,s)$ 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 $C_s^3$ 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 $n$…