Ramsey and Turán-type problems for non-crossing subgraphs of bipartite geometric graphs

Geometric versions of Ramsey-type and Turán-type problems are studied in a special but natural representation of bipartite graphs and similar questions are asked for general representations. A bipartite geometric graphG(m,n) = [A,B] is simple if the vertex classes A,B of G(m,n) are represented in R2 as A = {(1, 0), (2, 0), . . . , (m, 0)}, B = {(1, 1), (2, 1), . . . , (n, 1)} and the edge ab is the line segment joining a ∈ A and b ∈ B in R2. This and similar representations (two-layer representations) are studied earlier, and from the point of view of edge crossings, this representation is equivalent to others already in the literature, for example to cyclic bipartite graphs or to ordered bipartite graphs and certainly almost all textbook figures represent bipartite graphs this way. Subgraphs paths, trees, double stars, matchings are called non-crossing if they do not contain edges with common interior point. The choice of these subgraphs are explained by the fact that connected components of non-crossing subgraphs of simple bipartite geometric graphs must be special trees (caterpillars). We concentrate on balanced bipartite graphs, where m = n. The maximum number of edges is determined in a simple bipartite geometric graph G(n, n) that does not contain • non-crossing matchings with k + 1 edges • matchings with k + 1 pairwise crossing edges • non-crossing trees with k + 1 vertices