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Sep 6 at 5:13 comment added Erel Segal-Halevi @WillSawin you are right - if I understand correctly, it gives a lower bound of about $n^2/4$, and the factor can be substantially improved using later results. I added it to the question.
Sep 6 at 5:12 comment added Erel Segal-Halevi @GerryMyerson right, fixed
Sep 6 at 5:12 history edited Erel Segal-Halevi CC BY-SA 4.0
added 12 characters in body
Sep 6 at 5:07 history edited Erel Segal-Halevi CC BY-SA 4.0
added 12 characters in body
Sep 6 at 3:25 comment added Gerry Myerson "Given an integer $n$, I want to prove the existence of a binary matrix ...." Presumably, an $n\times n$ binary matrix.
Sep 6 at 1:53 comment added Will Sawin Doesn't the last result together with Bertrand's postulate give a lower bound for primes?
Sep 5 at 20:50 history asked Erel Segal-Halevi CC BY-SA 4.0