On the number of H-free hypergraphs

Two central problems in extremal combinatorics are concerned with estimating the number $ex(n,H)$, the size of the largest $H$-free hypergraph on $n$ vertices, and the number $forb(n,H)$ of $H$-free hypergraph on $n$ vertices. While it is known that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ for $k$-uniform hypergraphs that are not $k$-partite, estimates for hypergraphs that are $k$-partite (or degenerate) are not nearly as tight. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many degenerate hypergraphs $H$, $forb(n, H) = 2^{O(ex(n,H))}$. However, there are few known instances of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. In this paper, we show that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds for a wide class of degenerate hypergraphs known as $2$-contractible hypertrees. This is the first known infinite family of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. As a corollary of our main results, we obtain a surprisingly sharp estimate of $forb(n,C^{(k)}_\ell)=2^{(\lfloor\frac{\ell-1}{2}\rfloor+o(1))\binom{n}{k-1}}$ for the $k$-uniform linear $\ell$-cycle, for all pairs $k\geq 5, \ell\geq 3$, thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all $k\geq 5, \ell\geq 3$. Our methods also lead to some related sharp results on the corresponding random Turan problem. As a key ingredient of our proofs, we develop a novel supersaturation variant of the delta systems method for set systems, which may be of independent interest.

💡 Research Summary

The paper addresses two central extremal combinatorial functions for a fixed k‑uniform hypergraph H on n vertices: the Turán number ex(n,H), the maximum number of edges in an H‑free hypergraph, and the enumeration function forb(n,H), the total number of labelled H‑free hypergraphs. For non‑k‑partite (non‑degenerate) hypergraphs it is classical that forb(n,H)=2^{(1+o(1))ex(n,H)}. For degenerate (k‑partite) hypergraphs only the weaker bound forb(n,H)=2^{O(ex(n,H))} was known in general, due to Ferber, McKinley and Samotij. The authors identify a broad infinite family of degenerate hypergraphs—2‑contractible k‑trees—for which the sharp asymptotic equality holds.

A k‑tree is defined recursively: a single edge is a tree; otherwise there is a leaf edge whose removal leaves a smaller tree and whose intersection with the rest of the hypergraph is exactly the intersection with a unique parent edge. A hypergraph is t‑contractible if each edge contains t vertices of degree one; deleting those vertices yields a (k‑t)‑uniform multihypergraph called the t‑contraction. The paper focuses on the case t=2. For any such H, the minimum size of a cross‑cut (a set intersecting each edge in exactly one vertex) is denoted σ(H). Earlier work of Füredi and Jiang showed that for any subgraph of a 2‑contractible k‑tree, ex(n,H)=(σ(H)−1+o(1))·\binom{n}{k‑1}.

The main theorem (Theorem 1.3) proves that for every 2‑contractible k‑tree H, \