Unavoidable induced subgraphs in large graphs with no homogeneous sets

TL;DR

This paper proves that large prime graphs necessarily contain certain complex induced subgraphs or their complements, revealing unavoidable structures in large graphs without homogeneous sets.

Contribution

It establishes a universal bound ensuring large prime graphs contain specific induced subgraphs or their complements, advancing understanding of graph structure without homogeneous sets.

Findings

Large prime graphs contain specific induced subgraphs or their complements.

Identifies six particular graphs that must appear in sufficiently large prime graphs.

Provides bounds on the size of graphs to guarantee the presence of these subgraphs.

Abstract

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from $K_{1,n}$ by subdividing every edge once, (2) the line graph of $K_{2,n}$, (3) the line graph of the graph in (1), (4) the half-graph of height $n$, (5) a prime graph induced by a chain of length $n$, (6) two particular graphs obtained from the half-graph of height $n$ by making one side a clique and adding one vertex.