|Discrete Mathematics and Optimization Seminar
Friday August 31st at 4.30pm
Title. On a geometric Ramsey-style problem.
Given a point set P in R^2, the visibility graph V(P) is the graph
with vertex set P and for which, given points p,q
of P, pq is an edge precisely if the line segment [p,q] contains no
points aside from p and q. In 2006, Kara,
Por and Wood, conjectured that for all pairs of positive integers
(k,c) there is a finite number v(k,c) such that
if |P|> v(k,c) and no
k points of P are collinear, then V(P) contains a clique of size c.
Despite the simplicity of this statement,
essentially no non-trivial bounds on v(k,c) are known. I will discuss
some recent work on the conjecture of
Kara et.al. This
is joint work with Cristina Fernandes, Yoshiharu Kohayakawa, Jos
Coelho de Pina, and Yoshiko Wakabayashi.