Ordered and Convex Geometric Trees with Linear Extremal Function

The extremal functions ex → ( n , F ) and ex ↻ ( 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 ex → ( n , F ) and ex ↻ ( 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 T of ordered trees with k edges, and show that ex → ( n , T ) = ( k - 1 ) n - k 2 for all n ≥ k + 1 when T ∈ T and ex → ( n , T ) = Ω ( n log n ) for T ∉ T . We also describe the family T ′ of the convex geometric trees with linear Turán number and show that for every convex geometric tree F ∉ T ′ , ex ↻ ( n , F ) = Ω ( n log log n ) .