>k-homogeneous infinite graphs

In this article we give an explicit classification for the countably infinite graphs $\mathcal{G}$ which are, for some $k$, $\geq$$k$-homogeneous. It turns out that a $\geq$$k-$homogeneous graph $\mathcal{M}$ is non-homogeneous if and only if it is either not $1-$homogeneous or not $2-$homogeneous, both cases which may be classified using ramsey theory.Comment: 14 pages, 2 figure

Unavoidable induced subgraphs in large graphs with no homogeneous sets

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