A collection of families (F₁, F₂ , ⋯ , Fₖ) ∈ P([n])ᵏ is cross-Sperner if there is no pair i = j for which some Fᵢ ∈ Fᵢ is comparable to some Fⱼ ∈ Fⱼ. Two natural measures of the 'size' of such a family are the sum ∑i = 1ᵏ |Fᵢ| and the product ∏i = 1ᵏ |Fᵢ|. 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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Behague et al. (2024) studied this question.
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