Exact forbidden subposet results using Chain decompositions of the Cycle

TL;DR

This paper presents a chain decomposition method for cyclic permutations to determine the maximum size of subset families avoiding certain forbidden subposets, extending known results and establishing new bounds.

Contribution

It introduces a novel chain decomposition technique for cycle intervals and extends forbidden subposet bounds to broader classes of posets.

Findings

Established bounds for butterfly-free families.

Extended bounds to posets containing the butterfly as a subposet.

Derived LYM-type inequalities for these families.

Abstract

We introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the size of the largest family of subsets of $[n]:= \{1,2,...,n\}$ not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities.