Prime sum graphs and the induced trees they contain

Abstract

In this paper we show that prime sum graphs on $n$ vertices -- which are graphs on vertex set $\{1,2,...,n\}$ where $ij$ is an edge when $i+j$ is prime -- contain all trees with at most $\exp( c \log n / \log\log n)$ vertices as induced subgraphs. We also prove some results for related graphs, and end with some unsolved problems.