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2022

Saturation for Small Antichains

Đanković, Irina, Ivan, Maria-Romina

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For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets.

The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal A_{k})$.
Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\text{sat}^*(n, \mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\leq 4$.
In this paper we prove this conjecture for $k=5$ and $k=6$.

Saturation for Small Antichains · Around

Built on

D. Gerbner, B. Keszegh, N. Lemons, C. Palmer, D. Pálvölgyi, and B. Patkós, Saturating sperner families , Graphs and Combinatorics 29
2013
Earlier work this paper cites.
M. Ferrara, B. Kay, L. Kramer, R. R. Martin, B. Reiniger, H. C. Smith, and E. T. Sullivan, The saturation number of induced subposets of the boolean lattice , Discrete Mathematics 340
2017
Earlier work this paper cites.

Similar

D. Gerbner and B. Patkós, Extremal finite set theory , CRC Press, 2018
2018
Cited alongside, same era.
R. R Martin, H. C. Smith, and S. Walker, Improved bounds for induced poset saturation , The Electronic Journal of Combinatorics 27
Original
2020
Cited alongside, same era.

Then

B. Keszegh, N. Lemons, R. R. Martin, D. Pálvölgyi, and B. Patkós, Induced and non-induced poset saturation problems , Journal of Combinatorial Theory, Series A 184
2021
Later among the works it cites.

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