A note on Ordered Ruzsa-Szemer\'edi graphs

Abstract

A recent breakthrough of Behnezhad and Ghafari [FOCS 2024] and subsequent work of Assadi, Khanna, and Kiss [SODA 2025] gave algorithms for the fully dynamic $(1-\varepsilon)$-approximate maximum matching problem whose runtimes are determined by a purely combinatorial quantity: the maximum density of Ordered Ruzsa-Szemer\'edi (ORS) graphs. We say a graph $G$ is an $(r,t)$-ORS graph if its edges can be partitioned into $t$ matchings $M_1,M_2, \ldots, M_t$ each of size $r$, such that for every $i$, $M_i$ is an induced matching in the subgraph $M_{i} \cup M_{i+1} \cup \cdots \cup M_t$. This is a relaxation of the extensively-studied notion of a Ruzsa-Szemer\'edi (RS) graph, the difference being that in an RS graph each $M_i$ must be an induced matching in $G$. In this note, we show that these two notions are roughly equivalent. Specifically, let $\mathrm{ORS}(n)$ be the largest $t$ such that there exists an $n$-vertex ORS-$(\Omega(n), t)$ graph, and define $\mathrm{RS}(n)$ analogously. We show that if $\mathrm{ORS}(n) \ge \Omega(n^c)$, then for any fixed $\delta > 0$, $\mathrm{RS}(n) \ge \Omega(n^{c(1-\delta)})$. This resolves a question of Behnezhad and Ghafari.