On Kramer–Mesner matrix partitioning conjecture

In 1977, Ganter and Teirlinck proved that any 2 t × 2 t matrix with 2 t nonzero elements can be partitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any m t × n t matrix with k t nonzero elements can be partitioned into m n submatrices of order t of which at most k contain nonzero elements. In 1995, Brualdi et al. showed that this conjecture is true if m = 2 , k ≤ 3 or k ≥ m n − 2 . They also found a counterexample of this conjecture for m = 4 , n = 4 , k = 6 and t = 2 . When t = 2 , Rho showed that this conjecture is true if k ≤ 5 . When t = 2 and m = 3 , we show that this conjecture is true if k = 6 or n ≤ 3 . As a result, we show that when t = 2 , this conjecture is true if k = m n − 3 also.