Abstract In 2006, Marcus and Tardos proved that if are cyclic orders on some subsets of a set of symbols such that the common elements of any two distinct orders and appear in reversed cyclic order in and , then . This result is tight up to the logarithmic factor and has since become an important tool in Discrete Geometry. In this paper, we improve this to the optimal bound . In fact, we prove the following more general result. We show that if are linear orders on some subsets of a set of symbols such that no three symbols appear in the same order in any two distinct linear orders, then . Using this result, we resolve several open problems in Discrete Geometry and Extremal Graph Theory as follows. We prove that every ‐vertex topological graph that does not contain a self‐crossing four‐cycle has edges. This resolves a problem of Marcus and Tardos from 2006. We show that pseudo‐circles in the plane can be cut into pseudo‐segments, which, in turn, implies new bounds on the number of point‐circle incidences and on other geometric problems. This goes back to a problem of Tamaki and Tokuyama from 1998 and improves several results in the area. We also prove that the edge‐ordered Turán number of the four‐cycle is . This gives the first example of an edge‐ordered graph whose Turán number is known to be for some , and answers a question of Gerbner, Methuku, Nagy, Pálvölgyi, Tardos, and Vizer. Using different methods, we determine the largest possible extremal number that an edge‐ordered forest of order chromatic number two can have. Kucheriya and Tardos showed that every such graph has extremal number at most , and conjectured that this can be improved to . We disprove their conjecture in a strong form by showing that for every , there exists an edge‐ordered tree of order chromatic number two whose extremal number is .
Janzer et al. (Wed,) studied this question.