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 $pgeq 3$, every $K_{p}$-free semi-algebraic graph on $n$ vertices with independence number $o(n)$ has at most $rac{1}{2}left(1 - rac{1}{lceil p/2ceil-1} + o(1) ight)n^2$ 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.
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机译：半代数图$ G =（V，E）$是其中顶点是$ mathbb {R} ^ d $中的点且边集$ E $由半代数关系定义的图$ V $上的恒定复杂度。在此注释中，我们建立以下Ramsey-Turán定理：对于每个整数$ p geq 3 $，在独立性为$ o（n）$的$ n $个顶点上，每个无$ K_ {p} $无半代数图最多具有$ frac {1} {2} left（1- frac {1} { lceil p / 2 rceil-1} + o（1） right）n ^ 2 $条边。在此，对半代数关系的复杂性的依赖隐藏在$ o（1）$项中。此外，我们证明了这个界限是紧密的。
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