A Ramsey-type theorem in the plane

Pavel Valtr (joint work with Jaroslav Nesetril) Department of Applied Mathematics Charles University Czech Republic email: valtr@CSPGUK11.bitnet Report B 94-03 January 94

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We show that for any finite set $P$ of points in the plane and for any integer $k\ge2$ there is a finite set $R=R(P,k)$ with the following property: For any $k$-colouring of $R$ there is a monochromatic set $\ppp,\ppp\subseteq R,$ such that $\ppp$ is combinatorially equivalent to the set $P$ and the convex hull of $\ppp$ contains no point of $R\setminus\ppp.$ We also consider related questions for colourings of $p$-element subsets of $R$ ($p>1$) and show that these analogues have negative solutions.