The minimum cardinality of maximal systems of rectangular islands

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. 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 ) = ⌊ ( m n + 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 .