
Complete bipartite subgraphs of perfect graphs
Problem Let
be a perfect graph on
vertices. Is it true that either
or
contains a complete bipartite subgraph with bipartition
so that
?






Every perfect graph on vertices either has a clique or an independent set of size
, so weakening the bound on
,
to
gives a true statement. Jacob Fox [F] has proved that every comparability graph
on
vertices has a complete bipartite subgraph of size
, and (up to the constant) this is best possible.
Bibliography
[F] J. Fox, A Bipartite Analogue of Dilworth’s Theorem, Order 23 (2006), 197-209.
* indicates original appearance(s) of problem.
Claimed to be solved
In this recent preprint on Paul Seymour's webpage: http://web.math.princeton.edu/~pds/papers/pure5/paper.pdf (not on ArXiv nor published yet)