Problem

Source: China South East Mathematical Olympiad2011

Tags: combinatorics proposed, combinatorics



12 points are located on a clock with the sme distance , numbers $1,2,3 , ... 12$ are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct $n$ convex quadrilaterals . They satisfies the following conditions: (1). the colours of vertex of every convex quadrilateral are different from each other . (2). for every 3 quadrilaterals among them , there exists a colour such that : the numbers on the 3 points painted into this colour are different from each other . Find the maximum $n$ .