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.
Chan, Shortest distance in modular hyperbola and least quadratic non-residue, Mathematika 62 (2016), 860--865
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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.