Ramsey-Turán Numbers for Semi-Algebraic Graphs

Abstract

A semi-algebraic graph $G = (V,E)$ is a graph where the vertices are points in $\mathbb{R}^d$, and the edge set $E$ is defined by a semi-algebraic relation of constant complexity on $V$. In this note, we establish the following Ramsey-Turán theorem: for every integer $p\geq 3$, every $K_{p}$-free semi-algebraic graph on $n$ vertices with independence number $o(n)$ has at most $\frac{1}{2}\left(1 - \frac{1}{\lceil p/2\rceil-1} + o(1) \right)n^2$ edges. Here, the dependence on the complexity of the semi-algebraic relation is hidden in the $o(1)$ term.  Moreover, we show that this bound is tight.