A note on Ordered Ruzsa-Szemerédi graphs

TL;DR

This note investigates the relationship between Ordered Ruzsa-Szemerédi graphs (ORS) and classical Ruzsa-Szemerédi graphs (RS) in the context of fully dynamic $(1-\varepsilon)$-approximate maximum matching. It proves a density-boosting relation: if ${ORS}(n, \varepsilon n) \ge \Omega(n^{c})$ for some $c>0$, then for any fixed $\delta>0$, ${RS}(n, \Theta(\varepsilon^{1/\delta} n)) \ge \Omega(n^{c(1-\delta)})$, showing ORS and RS densities are roughly equivalent. The key technique is a tensor-power construction: from an ORS graph $G=\bigcup_i M_i$, build a graph $H_s$ on $V(G)^k$ and partition its edges into $M_a$ so that each $M_a$ is an induced matching, with the number of $M_a$'s giving the RS bound. This result implies that improvements in ORS-based dynamic-matching algorithms have parallel implications for RS-based bounds, resolving a question by Behnezhad and Ghafari about the relative power of ORS versus RS densities.

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-Szemerédi (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-Szemerédi (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-$(Ω(n), t)$ graph, and define $\mathrm{RS}(n)$ analogously. We show that if $\mathrm{ORS}(n) \ge Ω(n^c)$, then for any fixed $δ> 0$, $\mathrm{RS}(n) \ge Ω(n^{c(1-δ)})$. This resolves a question of Behnezhad and Ghafari.

Theorems & Definitions (6)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• proof : Proof of \ref{['main']}