Several New Generalizations of LYM Inequality
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 -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 -decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, -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 -decompositions to -multichains, and establish analogous LYM inequalities.
Cite
@article{arxiv.2509.21024,
title = {Several New Generalizations of LYM Inequality},
author = {Zihao Huang and Weikang Liang and Yujiao Ma and Suijie Wang},
journal= {arXiv preprint arXiv:2509.21024},
year = {2026}
}