On edge-ordered graphs with linear extremal functions

The systematic study of Turán-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. in 2020. Here we characterize connected edge-ordered graphs with linear extremal functions and show that the extremal function of other connected edge-ordered graphs is $\mathrm{\Omega }(n\log n)$. This characterization and dichotomy are similar in spirit to results of Füredi et al. (2020) about vertex-ordered and convex geometric graphs. We also extend the study of extremal function of short edge-ordered paths by Gerbner et al. to some longer paths.