Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron - Archive ouverte HAL
Article Dans Une Revue Combinatorial Theory Année : 2024

Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron

Carla Groenland
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  • PersonId : 1421863
Hugo Jacob
Tom Johnston
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  • PersonId : 1421865

Résumé

For given positive integers k and n, a family F of subsets of {1, . . . , n} is kantichain saturated if it does not contain an antichain of size k, but adding any set to F creates an antichain of size k. We use sat * (n, k) to denote the smallest size of such a family. For all k and sufficiently large n, we determine the exact value of sat * (n, k). Our result implies that sat * (n, k) = n(k -1) -Θ(k log k), which confirms several conjectures on antichain saturation. Previously, exact values for sat * (n, k) were only known for k up to 6.

We also prove a strengthening of a result of Lehman-Ron which may be of independent interest. We show that given m disjoint chains C 1 , . . . , C m in the Boolean lattice, we can create m disjoint skipless chains that cover the elements from ∪ m i=1 C i (where we call a chain skipless if any two consecutive elements differ in size by exactly one).

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Dates et versions

hal-04720670 , version 1 (03-10-2024)

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Paul Bastide, Carla Groenland, Hugo Jacob, Tom Johnston. Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron. Combinatorial Theory, 2024, 4 (1), ⟨10.5070/C64163848⟩. ⟨hal-04720670⟩
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