Avoiding \((m,m,m)\)-arrays of order \(n=2^k\) (Q426839)

Summary: An \((m,m,m)\)-array of order \(n\) is an \(n\times n\) array such that each cell is assigned a set of at most \(m\) symbols from \(\left\{1,\dots ,n\right\}\) such that no symbol occurs more than \(m\) times in any row or column. An \( (m,m,m)\)-array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant \(\gamma \) such that if \(m\leq\gamma 2^k\) and \(k\geq14\), then any \((m,m,m)\)-array of order \(n=2^k\) is avoidable. Such a constant \(\gamma\) has been conjectured to exist for all \(n\) by \textit{R. Häggkvist} [Discrete Math. 75, No.1--3, 253--254 (1989; Zbl 0669.05015)].