Table of Contents
Journal of Discrete Mathematics
Volume 2014 (2014), Article ID 731519, 5 pages
http://dx.doi.org/10.1155/2014/731519
Research Article

Noncrossing Monochromatic Subtrees and Staircases in 0-1 Matrices

1Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA
2Department of Mathematics, Pomona College, 640 North College Avenue, Claremont, CA 91711, USA
3Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O. Box 127, Budapest 1364, Hungary
4Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

Received 16 September 2013; Accepted 12 November 2013; Published 23 January 2014

Academic Editor: Wai Chee Shiu

Copyright © 2014 Siyuan Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The following question is asked by the senior author (Gyárfás (2011)). What is the order of the largest monochromatic noncrossing subtree (caterpillar) that exists in every 2-coloring of the edges of a simple geometric ? We solve one particular problem asked by Gyárfás (2011): separate the Ramsey number of noncrossing trees from the Ramsey number of noncrossing double stars. We also reformulate the question as a Ramsey-type problem for 0-1 matrices and pose the following conjecture. Every 0-1 matrix contains zeros or ones, forming a staircase: a sequence which goes right in rows and down in columns, possibly skipping elements, but not at turning points. We prove this conjecture in some special cases and put forward some related problems as well.