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–Károlyi–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 Turán number and show that for every convex geometric tree $F\notin {{\mathcal {T}}}'$, $\mathrm{{ex}}_{\circlearrowright }(n,F)= \Omega (n\log \log n)$.