Proof of Fishburn's latent-subset conjecture

Abstract

Fishburn's latent-subset conjecture, proposed in 1987 and revisited in 1988, asserts that for every dual intersecting family $\mathcal{F}\subseteq 2^{[n]}$, there exists $i\in [n]$ such that $|{\mathcal{F}}^L(i)|\ge |\mathcal{F}(i)|$. Here ${\mathcal{F}}^L$ is the family of the subsets of members of $\mathcal{F}$ that do not belong to $\mathcal{F}$. In this paper, we prove the conjecture using the recent weighted star inequality of Chang, Liu, and Liu.

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