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arXiv · 2509.21024

Several New Generalizations of LYM Inequality

Zihao Huang·Weikang Liang·Yujiao Ma·Suijie Wang
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Abstract

The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of $r$-decompositions, where all components are required to avoid chains of the same length. In this paper, we relax this constraint by allowing components of a family of $r$-decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, $q$-analog, continuous analog, and arithmetic analog frameworks. Notably, the bound in our LYM inequalities does not depend on the maximal length of all forbidden chains. Moreover, we extend our approach beyond $r$-decompositions to $r$-multichains, and establish analogous LYM inequalities.

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Zihao Huang, Weikang Liang, Yujiao Ma, Suijie Wang. 2025-09-25. Several New Generalizations of LYM Inequality. https://arxiv.org/abs/2509.21024

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