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 \(n\log ^{O(1)}n\) also holds for the same set of extremal functions.

Gerbner 等人发起了对边有序图的 Turán 型极值问题的系统研究。 （边序图的 Turán 问题，2021）。他们推测色数为 2 的边序森林的极值函数为\(n^{1+o(1)}\)。在这里，我们解决了这个猜想，证明了\(n2^{O(\sqrt{\log n})}\)的更强上限。这代表了可能的极值函数族中的一个差距，因为其他禁止的边序图对于某些\(c>1\)具有极值函数\(\Omega (n^c)\)。然而，我们的结果可能不是最终的结论：这里我们推测\(n\log ^{O(1)}n\)的更强上限也适用于同一组极值函数。