Abstract
The systematic study of Turán-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. (Turán 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(sqrt{log n})}. This represents a gap in the family of possible extremal functions as other forbidden edge-ordered graphs have extremal functions Omega (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 nlog ^{O(1)}n also holds for the same set of extremal functions.
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