 
Summary: Explicit Ramsey graphs and orthonormal labelings
Noga Alon
Abstract
We describe an explicit construction of trianglefree graphs with no independent sets of size
m and with (m3/2
) vertices, improving a sequence of previous constructions by various authors.
As a byproduct we show that the maximum possible value of the Lov´asz function of a graph
on n vertices with no independent set of size 3 is (n1/3
), slightly improving a result of Kashin
and Konyagin who showed that this maximum is at least (n1/3
/ log n) and at most O(n1/3
).
Our results imply that the maximum possible Euclidean norm of a sum of n unit vectors in Rn
,
so that among any three of them some two are orthogonal, is (n2/3
).
1 Introduction
Let R(3, m) denote the maximum number of vertices of a trianglefree graph whose independence
number is at most m. The problem of determining or estimating R(3,m) is a well studied Ramsey
type problem. Ajtai, Koml´os and Szemer´edi proved in [1] that R(3, m) O(m2/ log m), (see also
