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

n edge-ordered graph is a graph with a total ordering of its edges. A path P=v_1v_2… v_k in an edge-ordered graph is called increasing if (v_iv_i+1) < (v_i+1v_i+2) for all i = 1,…,k-2 ; and it is called decreasing if (v_iv_i+1) > (v_i+1v_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(loglog n) , α(K_n) = O(log n), τ(K_n) = Ω(logloglog n) and τ(K_n) = O(√(n log n)) .