Tur\'{a}n results for posets and their alternating cycles

Abstract

For a partially ordered set ${\mathbb{P}} = (X,\leq)$ there exist hypergraphs where the vertices are the set of ordered tuples of either all incomparable elements of ${\mathbb{P}}$ or all the critical pairs of ${\mathbb{P}}$, and the edges are formed by the duals of either all the alternating cycles of ${\mathbb{P}}$ or all the strict alternating cycles of ${\mathbb{P}}$. The weak chromatic numbers of these hypergraphs are all equal to the order dimension of ${\mathbb{P}}$. Here are established upper bounds on the number of strict alternating cycles a poset ${\mathbb{P}}=(X,\leq)$ can have in terms of $n = |X|$, the cardinality of the groundset of ${\mathbb{P}}$, and the width $w$ of ${\mathbb{P}}$. These bounds also apply to the number of hyperedges of the associated hypergraph ${\mathcal{H}}^s(\mathbb{P})$, with incomparable pairs as vertices and strict alternating cycles dual to its hyperedges.