 
Summary: Ramsey Graphs Cannot
Be Defined by Real
PolynomiaIs
Noga Alon
BELLCORE
MORRISTOWN, NEW JERSEY USA
DEPARTMENT OF MATHEMATICS
SACKLERFACULTY OF EXACTSCIENCES
TEL AVIV UNIVERSITL: TELAWL( ISRAEL
ABSTRACT
Let P(x,y,n)be a real polynomialand let {G, } be a family of graphs, where
the set of vertices of G, is (1.2,. . .,n}and for 1 Ii < j' 5 n {;,I] is an
edge of G, iff P(i,j,n)> 0. Motivated by a question of Babai, we show
that there is a positiye constant c depending only on P such that either G
or its complement G, contains _a complete subgraph on at least c2
vertices. Similarly, either G, or G, contains a complete bipartite subgraph
with at least cntnvertices in each color class. Similar results are proved for
graphs defined by real polynomials in a more general way, showing that
such graphs satisfy much stronger Ramsey bounds than do random
graphs. This may partially explain the difficulties in finding an explicit con
