Sous Les Pavés

Various combinatorial properties of Euclidean boxes are investigated. A box is defined as a product of intervals of the real line. Every intersection of boxes is a box, so boxes define a convexity structure whose usual parameters, Radon, Caratheodory, Helly and exchange numbers, have been evaluated by various authors. Related to the Erdos-Szekeres theorem on monotonic sequences, we prove the following: Among 2 2n-1 +1 points in R n , there exist three of them x, y, z such that x is ‘between’ y and z : every box containing y and z must contain x .