Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications

Abstract

In 2006, Marcus and Tardos proved that if $A^1,\dots,A^n$ are cyclic orders on some subsets of a set of $n$ symbols such that the common elements of any two distinct orders $A^i$ and $A^j$ appear in reversed cyclic order in $A^i$ and $A^j$, then $\sum_{i} |A^i|=O(n^{3/2}\log n)$. This result is tight up to the logarithmic factor and has since become an important tool in Discrete Geometry. We improve this to the optimal bound $O(n^{3/2})$. In fact, we show that if $A^1,\dots,A^n$ are linear orders on some subsets of a set of $n$ symbols such that no three symbols appear in the same order in any two distinct linear orders, then $\sum_{i} |A^i|=O(n^{3/2})$. Using this result, we resolve several open problems in Discrete Geometry and Extremal Graph Theory as follows. We prove that every $n$-vertex topological graph that does not contain a self-crossing four-cycle has $O(n^{3/2})$ edges. This resolves a problem of Marcus and Tardos from 2006. We also show that $n$ pseudo-circles in the plane can be cut into $O(n^{3/2})$ pseudo-segments, which, in turn, implies new bounds on point-circle incidences and on other geometric problems. Moreover, we prove that the edge-ordered Tur\'an number of the four-cycle $C_4^{1243}$ is $\Theta(n^{3/2})$. This answers a question of Gerbner, Methuku, Nagy, P\'alv\"olgyi, 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 $n2^{O(\sqrt{\log n})}$, and conjectured that this can be improved to $n(\log n)^{O(1)}$. We disprove their conjecture by showing that for every $C>0$, there exists an edge-ordered tree of order chromatic number two whose extremal number is $\Omega(n 2^{C\sqrt{\log n}})$.