A note on Ordered Ruzsa-Szemerédi graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909476127244288 |
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| author | Pratt, Kevin |
| author_facet | Pratt, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on Ordered Ruzsa-Szemerédi graphs Pratt, Kevin Data Structures and Algorithms Combinatorics 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. |
| title | A note on Ordered Ruzsa-Szemerédi graphs |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2502.02455 |