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

An edge-ordered graph is a with total ordering of its edges. A path $$P=v_1v_2\ldots v_k$$ in an called increasing if $$(v_iv_{i+1}) < (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ ; and it decreasing > . We say that P monotone or decreasing. rooted tree T either every from the root to leaf Let G be graph. In straight-line drawing D G, vertices are drawn as different points plane edges straight line segments. $$\overline{\alpha}(G)$$ largest integer such contains non-crossing length $$\overline{\tau}(G)$$ complete binary this paper we show $$\overline \alpha(K_n) \Omega(\log\log n)$$ , O(\log n), \overline \tau(K_n) \log O(\sqrt{n n})$$