Ramsey Graphs and Block Designs. I

Abstract

This paper establishes a connection between a certain class of Ramsey numbers for graphs and the class of symmetric block designs admitting a polarity. The main case considered here relates the projective planes over Galois fields to the Ramsey numbers $R({C_4},{K_{1,n}}) = f(n)$. It is shown that, for every prime power $q,f({q^2} + 1) = {q^2} + q + 2$, and $f({q^2}) = {q^2} + q + 1$.