An Extremal Problem for Finite Lattices

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For a positive integer k, a k-square Sk (more generally, a square) in Z×Z is any set {(i, j), (i+ k, j), (i, j + k), (i+ k, j + k)} ⊂ Z×Z. Let Sk denote the class of k-squares Sk ⊂ Z× Z. A set A ⊂ Z× Z is said to be Sk-free if, for each Sk ∈ Sk, we have that Sk 6⊆ A. For positive integers M and N , let LM,N = [0,M − 1] × [0, N − 1] be the M ×N non-negative integer lattice. For positive integers k1, . . . , k`, set ex(LM,N , {Sk1 , . . . ,Sk`}) = max {|A| : A ⊆ LM,N is Ski-free for all 1 ≤ i ≤ `} , and when {Sk1 , . . . ,Sk`} = {Sk}, we abbreviate this parameter to ex(LM,N ,Sk). Our first result gives an exact formula for ex(LM,N ,Sk) for all integers k,M,N ≥ 1, where ex(LM,N ,Sk) = (3/4 + o(1))MN holds for fixed k and min{M,N} → ∞. Our second result identifies a set A0 ⊂ LM,N of size |A0| ≥ (2/3)MN with the property that, for any integer k 6≡ 0 (mod 3), A0 is Sk-free. Our third result shows that |A0| is asymptotically best possible, in that for all integers M,N ≥ 1, ex(LM,N , {S1,S2}) ≤ (2/3)MN +O(M +N). When M = 3m is divisible by three, our estimates on the error O(M + N) render exact formulas for ex(L3m,3, {S1,S2}) and ex(L3m,6, {S1,S2}).