The Hajnal--Rothschild problem

2025-02-10

AI Summary

This paper resolves the Hajnal–Rothschild problem: determining the maximum size of a family $mathcal{F} subseteq inom{[n]}{k}$ such that among any $s+1$ sets in $mathcal{F}$, some pair has intersection size at least $t$. The authors introduce the **iterative propagation approximation technique**, extending the spread method to approximate multi-block structures. Under the condition $n > 2k + C t^{4/5} s^{1/5}(k-t)log^4 n$, they prove that extremal families are precisely unions of $s$ pairwise disjoint “shifted $t$-intersecting families”, i.e., $s$-fold compositions of the Complete $t$-Intersection Theorem. This fully characterizes the extremal structure, substantially improving upon Hajnal and Rothschild’s 1973 asymptotic bound and constructive results. The work represents a major advance in extremal set theory, providing the first precise structural description for multi-block extremal configurations in this classical problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Security and Privacy: Large-scale security measurements

Abstract

For a family $mathcal F$ define $ u(mathcal F,t)$ as the largest $s$ for which there exist $A_1,ldots, A_{s}in mathcal F$ such that for $i e j$ we have $|A_icap A_j|<t$. What is the largest family $mathcal Fsubset{[n]choose k}$ with $ u(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 $ u(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 $Ain mathcal F$ there is $iin [s]$ such that $|Acap X_i|ge t+x_i$. 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.

Problem

• Determine largest family with limited intersection size.
• Analyze structure of extremal families in combinatorics.
• Enhance spread approximation technique for proofs.

Innovation

• Enhanced spread approximation technique
• Iterative spread approximation method
• Complete t-Intersection Theorem application