Ordered and Convex Geometric Trees with Linear Extremal Function

Abstract

The extremal functions $$\mathrm{{ex}}_{\rightarrow }(n,F)$$ and $$\mathrm{{ex}}_{\circlearrowright }(n,F)$$ for ordered and convex geometric acyclic graphs F have been extensively investigated by a number of researchers. Basic questions are to determine when $$\mathrm{{ex}}_{\rightarrow }(n,F)$$ and $$\mathrm{{ex}}_{\circlearrowright }(n,F)$$ are linear in n, the latter posed by Brass–Karolyi–Valtr in 2003. In this paper, we answer both these questions for every tree F. We give a forbidden subgraph characterization for a family $${\mathcal {T}}$$ of ordered trees with k edges, and show that $$\mathrm{{ex}}_{\rightarrow }(n,T) = (k - 1)n - {k \atopwithdelims ()2}$$ for all $$n \ge k + 1$$ when $$T \in {{\mathcal {T}}}$$ and $$\mathrm{{ex}}_{\rightarrow }(n,T) = \Omega (n\log n)$$ for $$T \not \in {{\mathcal {T}}}$$ . We also describe the family $${{\mathcal {T}}}'$$ of the convex geometric trees with linear Turan number and show that for every convex geometric tree $$F\notin {{\mathcal {T}}}'$$ , $$\mathrm{{ex}}_{\circlearrowright }(n,F)= \Omega (n\log \log n)$$ .