f(n²,n) ≥ n + (1/√2 + o(1))√n and f(p²,p) ≤ 2p − (√(2/3) − o(1))√(p/log p) for large primes p.
Xu,A generalization of sets without long arithmetic progressions based on Szekeres algo- rithm, J
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Hitting Arithmetic Progressions at the Square-Root Scale
f(n²,n) ≥ n + (1/√2 + o(1))√n and f(p²,p) ≤ 2p − (√(2/3) − o(1))√(p/log p) for large primes p.