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arXiv · 2607.03026

Counterexamples to the Balogh-Linz-Patk\'os Conjecture

Jia-Bao Yang·Leilei Zhang
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Abstract

A set system $\mathcal{F}$ is called $t$-intersecting if $|A\cap B|\ge t$ for every pair of sets $A,B\in \mathcal{F}.$ A set system $\mathcal{F}$ is $k$-Sperner if it does not contain a chain of length $k+1$. Balogh, Linz and Patk\'os (Combinatorial Theory, 2023) conjectured an extremal result for $t$-intersecting $k$-Sperner families when $n+t$ is odd. In this note we give an explicit construction that is $t$-intersecting and $k$-Sperner, and whose size exceeds that of the conjectured fixed-star construction for infinitely many values of $n$. Consequently, we disprove the Balogh-Linz-Patk\'os conjecture for all $t\ge 2$ and $k\ge 2$ satisfying $k(t-1)\ge t+1$.

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Jia-Bao Yang, Leilei Zhang. 2026-07-03. Counterexamples to the Balogh-Linz-Patk\'os Conjecture. https://arxiv.org/abs/2607.03026

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