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Few exact values of $R(B_n,B_n)$ have been obtained since Rousseau and Sheehan (1978) proved, using Paley graphs, $R(B_n, B_n) = 4n + 2$ whenever $4n+1$ is a prime power.\n  In this paper, we obtain $R(B_n,B_n)=4n+1$ for infinitely many $n$ by constructing new families of strongly regular graphs. Moreover, we prove that $R(B_{n-2},B_n)\\le 4n-3$ for every $n\\ge 3$ with $n\\ne 6$, removing the original condition $n\\equiv 2\\pmod 3$ due to Rousseau and Sheehan. 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