Let $C_{k}^{(r)}$ be the $r$-uniform linear cycle on $k$ hyperedges. An $r$-graph is $C_{k}^{(r)}$-free if it contains no copy of $C_{k}^{(r)}$. Let $\operatorname{ex}_{r}(n,C_{k}^{(r)})$ denote the maximum number of hyperedges in an $n$-vertex $C_{k}^{(r)}$-free $r$-graph. Balogh, Narayanan and Skokan asked whether, for every pair of integers $r,k\ge 3$, the number of $C_k^{(r)}$-free $r$-graphs on $n$ labelled vertices is \[ 2^{(1+o(1))\operatorname{ex}_{r}(n,C_{k}^{(r)})}. \] While the analogous statement is known to fail for graphs ($r=2$), by a construction of Morris and Saxton, the general question remained open for hypergraphs. Very recently, Jiang and Longbrake answered the question affirmatively when $r\geq5$. In this paper, we completely resolve the problem by proving that the answer is affirmative for every pair of integers $r,k\geq3$. When $(r,k)\neq(3,3)$, we establish balanced supersaturation results for linear cycles and combine them with the hypergraph container method. For $(r,k)=(3,3)$, we instead decompose $3$-graphs according to their pair-codegrees, encode the subhypergraph formed by the hyperedges that contain a large codegree pair as a directed graph, and apply a multicolour entropy theorem.
For a family $\mathcal{F}$ of $k$-graphs, $ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n\g...
Let $\delta^*_{k,r}$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\ge2$, $r\ge k+1$ and $\varepsilon>0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $(\delta^*_{k,r}+\varepsilon)n$ admits, with high probability, a fractional $K_r^{(k...
Let $u_k^{(r)}(H,p)$ be the maximum, over all $k$-vertex $r$-uniform hypergraphs $F$, of the absolute difference between the number of induced copies of $F$ in $H$ and its expectation in the binomial random $r$-graph with the same number of vertices as $H$ and edge probability $p$. Let $u_k^{(r)}(n,p)$ be the minimum o...
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)...
For an $r$-uniform hypergraph $\mathcal H$, let $\ex_r(n,\mathcal H)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no copy of $\mathcal H$. The $r$-uniform expansion $T^{(r)}$ of a tree $T$ is obtained by adding $r-2$ new vertices to each edge, with the added sets pairwise disjo...
Let $H = (V, E)$ be a $k$-uniform hypergraph. For $1 ≤ s ≤ k − 1,$ an $s$-path $P^{(k,s)}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1, v_2, · · · , v_{s+n(k−s)}$ such that $\{v_{1+i(k-s)}, \cdots, v_{s+(i+1)(k-s)}\}$ is an edge of $H$ for each $0 ≤ i ≤ n−1.$ In this paper, we prove that $R(P^ {(3s,s...
Er-Xiong Liu· Annals of Applied Mathematic...· 0 citations
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