The minimum cardinality of maximal systems of rectangular islands

• Authors: Zsolt Lengvárszky
• Venue: European Journal of Combinatorics
• Year: 2009

For given positive integers m and n, and R={(x,y):0@?x@?m and 0@?y@?n}, a set H of rectangles that are all subsets of R and the vertices of which have integer coordinates is called a system of rectangular islands if for every pair of rectangles in H one of them contains the other or they do not overlap at all. Let I"R denote the ordered set of systems of rectangular islands on R, and let max(I"R) denote the maximal elements of I"R. For f(m,n)=max{|H|:H@?max(I"R)}, G. Czedli [G. Czedli, The number of rectangular islands by means of distributive lattices, European J. Combin. 30 (1) (2009) 208-215)] proved f(m,n)=@?(mn+m+n-1)/2@?. For g(m,n)=min{|H|:H@?max(I"R)}, we prove g(m,n)=m+n-1. We also show that for any integer h in the interval [g(m,n),f(m,n)], there exists an H@?max(I"R) such that |H|=h.