Abstract
Suppose k ≥ 2 is an integer. Let Yk be the poset with elements x1,x2,y1,y2,…,yk− 1 such that y1 < y2 < ⋯ < yk− 1 < x1,x2 and let \(Y_{k}^{\prime }\) be the same poset but all relations reversed. We say that a family of subsets of [n] contains a copy of Yk on consecutive levels if it contains k + 1 subsets F1,F2,G1,G2,…,Gk− 1 such that G1 ⊂ G2 ⊂⋯ ⊂ Gk− 1 ⊂ F1,F2 and |F1| = |F2| = |Gk− 1| + 1 = |Gk− 2| + 2 = ⋯ = |G1| + k − 1. If both Yk and \(Y^{\prime }_{k}\) on consecutive levels are forbidden, the size of the largest such family is denoted by \(\text {La}_{\mathrm {c}}\left (n, Y_{k}, Y^{\prime }_{k}\right )\). In this paper, we will determine the exact value of \(\text {La}_{\mathrm {c}}\left (n, Y_{k}, Y^{\prime }_{k}\right )\).
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Acknowledgements
We are indebted to the anonymous referees for their useful suggestions. The first author is partially supported by the National Research, Development and Innovation Office – NKFIH under the grant SSN117879, NK104183 and K116769. The second author is partially supported by the National Natural Science Foundation of China (No. 11671320).
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Katona, G.O.H., Xiao, J. Largest Family Without a Pair of Posets on Consecutive Levels of the Boolean Lattice. Order 39, 15–27 (2022). https://doi.org/10.1007/s11083-021-09558-3
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DOI: https://doi.org/10.1007/s11083-021-09558-3