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This is a little late, but since this is the first thing that pops up with a google search of cliques in Paley graphs, I will mention the following: here is a resultresult that improves the bound of $\sqrt{p}$ to $\sqrt{p} - 1$ for infinitely many primes.

This is a little late, but since this is the first thing that pops up with a google search of cliques in Paley graphs, I will mention the following: here is a result that improves the bound of $\sqrt{p}$ to $\sqrt{p} - 1$ for infinitely many primes.

This is a little late, but since this is the first thing that pops up with a google search of cliques in Paley graphs, I will mention the following: here is a result that improves the bound of $\sqrt{p}$ to $\sqrt{p} - 1$ for infinitely many primes.

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George Shakan
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This is a little late, but since this is the first thing that pops up with a google search of cliques in Paley graphs, I will mention the following: here is a result that improves the bound of $\sqrt{p}$ to $\sqrt{p} - 1$ for infinitely many primes.