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The Hajnal--Rothschild problem
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The Hajnal--Rothschild problem
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For a family $\mathcal F$ define $\nu(\mathcal F,t)$ 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 $\nu(\mathcal F,t)\le s$? 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 $\nu(\mathcal F,t)\le s$ 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.
Forward citations
Cited by 4 Pith papers
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