The size of maximal systems of square islands

For integers m > 0 , n > 0 , and R = { ( x , y ) : 0 ≤ x ≤ m  and  0 ≤ y ≤ n } , a set H of closed 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 Journal of Combinatorics, in press (doi:10.1016/j.ejc.2008.02.005)] proved f ( m , n ) = ⌊ ( m n + m + n − 1 ) / 2 ⌋ . For g ( m , n ) = min { | H | : H ∈ max ( I R ) } in [Z. Lengvárszky, The minimum cardinality of maximal systems of rectangular islands, European Journal of Combinatorics 30 (1) 216–219], we proved g ( m , n ) = m + n − 1 . Systems of square islands are systems of rectangular islands with R and all members of H being squares. The functions f ( n ) and g ( n ) are defined analogously to f ( m , n ) and g ( m , n ) , and we show f ( n ) ≤ n ( n + 2 ) / 3 (best polynomial bound), and g ( n ) = n .