Noncrossing Monochromatic Subtrees and Staircases in 0-1 Matrices

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 n−1 zeros or n−1 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