Families of Subsets Without a Given Poset in the Interval Chains

TL;DR

This paper establishes improved upper bounds for the size of P-free subposets within double chains, using graph independence numbers, and provides methods to construct posets satisfying the Griggs-Lu conjecture.

Contribution

It introduces a new upper bound for the maximum size of P-free subposets in double chains and offers polynomial-time methods to determine independence numbers of related graphs.

Findings

Improved upper bounds for $ extstyle ext{La}(Q,P)$ when $Q$ is a double chain.

Polynomial-time algorithms for finding independence numbers of auxiliary graphs.

Construction methods for posets satisfying the Griggs-Lu conjecture.

Abstract

For two posets $P$ and $Q$, we say $Q$ is $P$-free if there does not exist any order-preserving injection from $P$ to $Q$. The speical case for $Q$ being the Boolean lattice $B_n$ is well-studied, and the optiamal value is denoted as $\lanp$. Let us define $\La(Q,P)$ to be the largest size of any $P$-free subposet of $Q$. In this paper, we give an upper bound for $\La(Q,P)$ when $Q$ is a double chain and $P$ is any graded poset, which is better than the previous known upper bound, by means of finding the indpendence number of an auxiliary graph related to $P$. For the auxiliary graph, we can find its independence number in polynomial time. In addition, we give methods to construct the posets satisfying the Griggs-Lu conjecture.