The Hajnal--Rothschild problem
Peter Frankl, Andrey Kupavskii
TL;DR
This work resolves, in a broad asymptotic regime, the Hajnal–Rothschild problem for $k$-uniform hypergraphs by proving that the largest family with $\nu(\mathcal{F},t)\le s$ is structurally a union of $s$ $t$-intersecting cliques, i.e., there exist sets $X_1,\ldots,X_s$ with $|X_i|=t+2x_i$ such that every $A$ satisfies $|A\cap X_i|\ge t+x_i$ for some $i$, mirroring the Complete $t$-Intersection Theorem. The main technique, spread approximation, is enhanced with iterative spread and a peeling process to obtain a coarse-to-fine analysis: first a low-uniformity approximation $\,\mathcal{S}$ with $\nu(\mathcal{S},t)\le s$ and a small remainder, then a fine-grained structure theorem that identifies the extremal as a union of $t$-intersecting cliques, and finally a remainder-elimination step that yields the exact extremal form. The results hold under explicit bounds on $n$ in terms of $k,t,s$, notably $n>2k+C(k-t)t^{4/5}s^{1/5}\log^4 n$ and $n>2k+Cs(k-t)\log^4 n$, and the methods provide a robust framework for deriving stability and structure in large-n regimes. Overall, the paper extends Hajnal and Rothschild’s theorem with quasi-polynomial dependencies and reveals a stable, clique-based decomposition of near-extremal families, connecting to Ahlswede–Khachatrian-type classifications for union-structured extremals.
Abstract
For a family $\mathcal F$ define $ν(\mathcal F,t)$ as the largest $s$ for which there exist $A_1,\ldots, A_{s}\in \mathcal F$ such that for $i\ne j$ we have $|A_i\cap A_j|< t$. What is the largest family $\mathcal F\subset{[n]\choose k}$ with $ν(\mathcal F,t)\le s$? This question goes back to a paper Hajnal and Rothschild from 1973. We show that, for some absolute $C$ and $n>2k+Ct^{4/5}s^{1/5}(k-t)\log_2^4n$, $n>2k+Cs(k-t)\log_2^4 n$ the largest family with $ν(\mathcal F,t)\le s$ has the following structure: there are sets $X_1,\ldots, X_s$ of sizes $t+2x_1,\ldots, t+2x_s$, such that for any $A\in \mathcal F$ there is $i\in [s]$ such that $|A\cap X_i|\ge t+x_i$. That is, the extremal constructions are unions of the extremal constructions in the Complete $t$-Intersection Theorem. For the proof, we enhance the spread approximation technique of Zakharov and the second author. In particular, we introduce the idea of iterative spread approximation.