Minimising the total number of subsets and supersets

Gowty, Adam; Horsley, Daniel; Mammoliti, Adam

doi:10.48550/arxiv.2212.13112

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2023

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“…Note added before submission: In the final stages of preparation of this article, we noticed a recent paper of Gowty, Horsley, and Mammoliti [9], concerning the comparability number. They give a very different proof of Theorem 2.3 (see Corollary 1.2 of [9]) and use it as we do to deduce Theorem 2.4. They also provide some very interesting further analysis of the comparability number and sets that minimise c(n, m).…”

Section: Closing Remarksmentioning

confidence: 99%

Improved bounds for cross-Sperner systems

Natalie¹,

Kuperus²,

Morrison³

et al. 2023

Preprint

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A collection of families (Fthere is no pair i = j for which some F i ∈ F i is comparable to some F j ∈ F j . Two natural measures of the 'size' of such a family are the sum k i=1 |F i | and the product k i=1 |F i |. We prove new upper and lower bounds on both of these measures for general n and k ≥ 2 which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011.

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Section: Closing Remarksmentioning

confidence: 99%