Non-crossing monotone paths and binary trees in edge-ordered complete geometric graphs

An edge-ordered graph is a graph with a total ordering of its edges. A path P = v 1 v 2 … v k in an edge-ordered graph is called increasing if ( v i v i + 1 ) < ( v i + 1 v i + 2 ) for all i = 1 , … , k - 2 ; and it is called decreasing if ( v i v i + 1 ) > ( v i + 1 v i + 2 ) for all i = 1 , … , k - 2 . We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing. Let G be a graph. In a straight-line drawing D of G , its vertices are drawn as different points in the plane and its edges are straight line segments. Let α ¯ ( G ) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length α ¯ ( G ) . Let τ ¯ ( G ) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of τ ¯ ( G ) edges. In this paper we show that α ¯ ( K n ) = Ω ( log log n ) , α ¯ ( K n ) = O ( log n ) , τ ¯ ( K n ) = Ω ( log log log n ) and τ ¯ ( K n ) = O ( n log n ) .