A minimax theorem on intervals

Abstract

A problem of V. Chvátal is solved. It is proved that if P is a vertically convex lattice polygon with vertical and horizontal sides then the minimum number of the rectangles in P which cover P is equal to the maximum cardinality of the point sets such that any two elements induce a rectangle (with vertical and horizontal sides) not contained by the polygon P. This result is the best possible in some sense. Actually the following conjecture of A. Frank is proved which implies this theorem. A sequence of intervals I 1,…, I m is U-increasing if − u j=1 k−1 I j ≠ ∪ j=1 k I j for k = 2, 3,…, m. The set of intervals G is a generating set for the set of intervals J if every element of J is a union of members of G . We prove the conjecture that for any set of intervals J , the minimum size of a generating set G for J is equal to the maximum size of a U-increasing sequence of intervals with members from J .