A Characterization of Edge-Ordered Graphs with Almost Linear Extremal Functions

The systematic study of Turan-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. (Turan problems for Edge-ordered graphs, 2021). They conjectured that the extremal functions of edge-ordered forests of order chromatic number 2 are n(1+o(1)). Here we resolve this conjecture proving the stronger upper bound of n2(O(vlog n)). This represents a gap in the family of possible extremal functions as other forbidden edge-ordered graphs have extremal functions O(n(C)) for some c > 1. However, our result is probably not the last word: here we conjecture that the even stronger upper bound of n log(O(1)) n also holds for the same set of extremal functions.