For infinitely many primes p there is a choice of c such that no square of side sqrt(0.1 log p) contains two points, giving a sqrt(log p) lower bound for the guaranteed side length.
Karatsuba, Distribution of values of Dirichlet characters on additive sequences, Soviet Math
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Close points on a modular hyperbola
For infinitely many primes p there is a choice of c such that no square of side sqrt(0.1 log p) contains two points, giving a sqrt(log p) lower bound for the guaranteed side length.