Sperner's problem for G-independent families

TL;DR

This paper investigates the size of the largest antichain in the independent set lattice of a path graph, extending Sperner's theorem to a specific graph class and establishing bounds comparable to the largest layer.

Contribution

The paper proves that for a path graph, the maximum antichain size is asymptotically similar to the largest layer of the independent set lattice, extending Sperner's theorem.

Findings

Maximal antichain size is of the same order as the largest layer in Q(G) for path graphs.

Provides bounds and asymptotic behavior for antichains in G-independent families.

Extends Sperner's theorem to a new class of graphs.

Abstract

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by Sperner's Theorem. In this paper, we focus on the case where G is the path of length n-1, proving the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).