Matchings avoiding ordered patterns

TL;DR

This work resolves two central open Turán-type questions for ordered graphs: the exact maximum number of edges avoiding a non-separated k-matching, and tight bounds for avoiding a non-nested k-matching, plus exact results for alternating paths. The authors develop shifting arguments and edge-decomposition techniques to derive exact formulas and tight bounds, and connect these extremal results to Ramsey-type problems in ordered graphs. They also analyze the interplay between pattern classes (crossing, nested, separated) and reveal nuanced, non-intuitive phenomena in interval chromatic number behavior, including constructions showing separations between seemingly related pattern families. The results advance the ordered-graph Turán framework, with concrete implications for ordered Ramsey numbers and several open questions on forbidding multiple patterns simultaneously.

Abstract

A {\it vertex-ordered} graph is a graph equipped with a linear ordering of its vertices. A pair of independent edges in an ordered graph can exhibit one of the following three patterns: separated, nested or crossing. We say a pair of independent edges is non-separated if it is either crossing or nested. Non-nested and non-crossing pairs are defined analogously. We are interested in the following Turán-type problems: for each of the aforementioned six patterns, determine the maximum number of edges of an $n$-vertex ordered graph that does not contain a $k$-matching such that every pair of edges exhibit the fixed pattern. Exact answers have already been obtained for four of the six cases. The main objective of this paper is to investigate the two remaining open cases, namely non-separated and non-nested matchings. We determine the exact maximum number of edges of an $n$-vertex ordered graph that does not contain a non-separated $k$-matching, which has the form $\frac{3}{2}(k-1)n+Θ(k^2)$. For the non-nested case, we show the maximum number of edges lies between $(k-1)n$ and $(k-1)n+\binom{k-1}{2}$. We also determine the exact maximum number of edges of an $n$-vertex ordered graph that does not contain an alternating path of given length. We discuss some related problems and raise several conjectures. Furthermore, our results and conjectures yield consequences to certain Ramsey-type problems for non-nested matchings and alternating paths.

Figures (6)

• Figure 1: From right to left: separated, nested and crossing pairs.
• Figure 2: An extremal construction for Theorem \ref{['thm:nonsep']} with $n=14$ and $k=3$.
• Figure 3: A typical strongly non-nested matching.
• Figure 4: A construction achieving the lower bound of Theorem \ref{['thm:non-nested']} with $k=3$, obtained by starting with a $5$-vertex clique and repeatedly applying step 1.
• Figure 5: A construction achieving the lower bound of Theorem \ref{['thm:non-nested']} with $k=3$ and vertex set $[12]$. An edge $xy$ is included if either $x$ and/or $y$ is red, or if $|x-y|\in\{10,11\}$.

Theorems & Definitions (32)

• Definition 1.1
• Theorem 1.2: Pach and Tardos PT06
• Definition 1.3: Separated, nested and crossing edges
• Definition 1.4: $k$-matchings with special patterns
• Theorem 1.5: Kupitz Kupitz
• Theorem 1.6: Capoyleas and Pach CapoyleasP
• Theorem 1.7: Dujmović and Wood layout
• Theorem 1.8: Dujmović and Wood layout
• Theorem 2.1
• proof