The minimum cardinality of maximal systems of rectangular islands

For given positive integers mm and nn, and R={(x,y):0≤x≤m and 0≤y≤n}R={(x,y):0≤x≤m and 0≤y≤n}, a set HH of rectangles that are all subsets of RR and the vertices of which have integer coordinates is called a system of rectangular islands if for every pair of rectangles in HH one of them contains the other or they do not overlap at all. Let IRIR denote the ordered set of systems of rectangular islands on RR, and let max(IR)max(IR) denote the maximal elements of IRIR. For f(m,n)=max{|H|:H∈max(IR)}f(m,n)=max{|H|:H∈max(IR)}, G. Czédli [G. Czédli, 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⌋f(m,n)=⌊(mn+m+n−1)/2⌋. For g(m,n)=min{|H|:H∈max(IR)}g(m,n)=min{|H|:H∈max(IR)}, we prove g(m,n)=m+n−1g(m,n)=m+n−1. We also show that for any integer hh in the interval [g(m,n),f(m,n)][g(m,n),f(m,n)], there exists an H∈max(IR)H∈max(IR) such that |H|=h|H|=h.