Unavoidable induced subgraphs in large graphs with no homogeneous sets

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.Comment: 13 pages, 3 figure

On unavoidable‐induced subgraphs in large prime graphs

“…In this section, we state relevant definitions and notation, most of which, but not all, is from . Given a set X , let .…”

“…We think this outline is sufficient for understanding the global structure of the proof. For more details we direct the reader to the original article . Throughout denotes the smallest integer m such for that any coloring of the edges of with k , there is complete graph on vertices in color i for some .…”

“…Recently, Chudnovsky, Kim, Oum, and Seymour established that any prime graph contains one of a short list of induced prime subgraphs . A module of a graph is a set of vertices such that any vertex is either connected or nonconnected to all vertices in X .…”

“…In the present article, we reprove the main theorem of making use of model‐theoretic ingredients, in a way that improves the bounds and offers a different structural perspective on the graphs in question. Our background aim is to exemplify the usefulness of model‐theoretic ideas in proofs in finite combinatorics.…”

“…The model‐theoretic contribution of the present argument may be described as follows. The proof of Theorem 1.2 in proceeds by means of several cases, sketched in Section below, and applies Ramsey's theorem as a main tool. In it was shown that Ramsey's theorem works much better when the graph is so‐called stable, a finitization of an important structural property identified by model theory (for history, see the introduction to or the original source ).…”