A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals

We prove a partitioned version of the Erdös–Szekeres theorem for the case $k = 4$: any finite set $X \subset \bbbr^2$ of points in general position can be partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$, so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$ every transversal $\{x_1,x_2,x_3,x_4\}$, $x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4 \in X_{4j}$, is in convex position. In order to prove this, we show another theorem, the partitioned version of the “same type lemma”, which was proved by Bárány and Valtr.