A New Approach to the Results of Kövari, Sós, and Turán concerning Rectangle-free Subsets of the Grid

For positive integers m and n, define f(m,n) to be the smallest integer such that any subset A of the m × n integer grid with |A| ≥ f(m,n) contains a rectangle; that is, there are x ∈ [m] and y ∈ [n] and d1, d2 ∈ Z+ such that all four points (x, y), (x + d1, y), (x, y + d2), and (x + d1, y + d2) are contained in A. In 1954, Kövari, Sós, and Turán showed that lim k→∞ f(k, k) k3/2 = 1. They also showed that f(p2, p2 + p) = p2(p + 1) + 1 whenever p is a prime number. We recover their asymptotic result and strengthen the second, providing cleaner proofs which exploit a connection to projective planes, first noticed by Mendelsohn. We also provide an explicit lower bound for f(k, k) which holds for all k.