The Extremal Number of Tight Cycles

Abstract

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell \geq r+1$ vertices $x_1,...,x_{\ell }$ such that all $r$-tuples $\{x_{i},x_{i+1},...,x_{i+r-1}\}$ (with subscripts modulo $\ell $) are edges of $\mathcal{H}$. An old problem of V. Sós, also posed independently by J. Verstraëte, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices, which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$. This is tight up to the $o(1)$ error term, and it was shown recently by B. Janzer that this error term is indeed needed. Our proof is based on finding robust expanders in the line-graph of $\mathcal{H}$ together with certain density increment type arguments.

Hypergraphs with no tight cycles

“…For r = 3, an unpublished result of Verstraëte regarding the extremal number of a tight cycle of length 24 implies that ex 3 (n, C) = O(n 5/2 ). A recent result of Tomon and Sudakov [12] shows that ex r (n, C) ≤ n r−1 e O( √ log n) , greatly improving on previous bounds, and thus establishing that ex r (n, C) = n r−1+o (1) .…”

Robust (rainbow) subdivisions and simplicial cycles

“…Given a 3-graph G, let p(G) denote the number of 1-dimensional faces of G, that is, the number of pairs of vertices which appear in an edge. As proved by Letzter [33], slightly improving the result of Sudakov and Tomon [42], if a 3-graph G satisfies e(G) = Ω(p(G)(log p(G)) 5 ), then G contains a tight cycle. A similar strengthening of Theorem 1.5 also holds.…”

“…Graphs with such expansion properties commonly appear in the study of sparse extremal problems, see e.g. [23,40,41,42]. Our definition immediately implies their existence in every graph with positive constant average degree, unlike earlier arguments.…”

“…This was disproved in a strong sense by B. Janzer [17], who showed that an r-graph on n vertices can have Ω(n r−1 log n/ log log n) edges without containing a tight cycle. On the other hand, Sudakov and Tomon [42] proved the upper bound n r−1+o (1) , which was further improved by Letzter [33] to O(n r−1 (log n) 5 ). However, none of these proofs seem to extend to give any bounds on the extremal numbers of tight cycle of fixed length ℓ. Verstraëte [43] proposed the conjecture that if r divides ℓ, then any r-graph with n vertices containing no tight cycle of length ℓ can have at most O(n r−1+(r−1)/ℓ ) edges.…”

Rainbow clique subdivisions and blow-ups

“…The method used in [5] utilises robust expanders in the coloured setting together with a density increment argument, inspired in part by the method introduced by Sudakov and Tomon [12].…”