TY - JOUR A2 - Shiu, Wai Chee AU - Cai, Siyuan AU - Grindstaff, Gillian AU - Gyárfás, András AU - Shull, Warren PY - 2014 DA - 2014/01/23 TI - Noncrossing Monochromatic Subtrees and Staircases in 0-1 Matrices SP - 731519 VL - 2014 AB - 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 Kn,n? 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 n×n 0-1 matrix contains n1 zeros or n1 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. SN - 2090-9837 UR - https://doi.org/10.1155/2014/731519 DO - 10.1155/2014/731519 JF - Journal of Discrete Mathematics PB - Hindawi Publishing Corporation KW - ER -