{"id":"85f99aa9-289c-4b02-996d-67f7fb4015c3","arxiv_id":"2502.02455","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum density of Ordered Ruzsa-Szemerédi graphs is polynomially equivalent to the maximum density of ordinary Ruzsa-Szemerédi graphs.","lead":"An Ordered Ruzsa-Szemerédi graph is a graph whose edges can be split into many matchings with an ordering constraint. This note proves that dense ordered graphs of this kind force dense ordinary Ruzsa-Szemerédi graphs, so the two notions are polynomially equivalent.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the ORS-to-RS tensor construction is sound, and the only blemish is a harmless constant-factor undercount of matching sizes.","rationale":"I read the proof of Lemma 2.1 and Theorem 1.3 in detail. The construction is a tensor-power argument: from an ORS graph G, build H_s on V(G)^k with edges summing to s, and partition its edges by the coordinate matching indices. The critical step is showing that each M_a is an induced matching in H_s. That step is sound: for any edge in the induced subgraph, the ORS condition on the suffix graph gives f(U_i,V_i) ≤ a_i for each coordinate, and since the sums are equal, equality holds coordinate-wise. The reader correctly identified this coordinate-wise bound as the mechanism that depends on the suffix-induced definition of ORS; the paper applies it correctly, so the identified assumption is not a weakness in the actual proof. The only real blemish is the size computation: |M_a| is 2^{k-1}r^k rather than r^k because each coordinate edge can be oriented in two ways and the global reversal identifies pairs. This is a constant-factor error in the wrong direction: it makes the matchings larger, not smaller, so the stated lower bound still follows by taking subgraphs. Thus the central claim is robust, and the reader's ACCEPT verdict is unchanged. I would recommend correcting the count in any revision, but it does not affect the theorem.","tokens_in":3736,"tokens_out":26381,"duration_ms":243285,"concrete_test":"Work out the k=2, r=1 case of Lemma 2.1: with a=(1,2), the tensor construction on one edge of M1 and one edge of M2 yields two unordered edges, not one. Then verify that selecting exactly one of these two edges from every M_a still gives an induced matching, so RS(n^2,1) ≥ t^{1/2} remains valid; this isolates the constant-factor slip from any asymptotic conclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.3 is supported. The load-bearing step in Lemma 2.1 is the coordinate-wise inequality f(U_i,V_i) ≤ a_i for an edge (U,V) in the subgraph induced by M_a; this follows correctly from the suffix-induced ORS property, and the equality of sums then forces f(U_i,V_i) = a_i for every i. The induced-matching argument is therefore valid. The one concrete slip is the counting of |M_a|: for an undirected tensor graph, the construction gives (2r)^k ordered coordinate tuples and each unordered edge is counted twice, so the true size is 2^{k-1}r^k, not r^k as claimed on page 3. This only makes every M_a larger than stated. Since one may delete excess edges from each M_a and inducedness is preserved for any subset of an induced matching, the stated bound RS(n^k, r^k) ≥ t^{k-1}/k and the asymptotic theorem remain correct. No load-bearing concern remains.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This note proves that Ordered Ruzsa-Szemerédi (ORS) graphs and classical Ruzsa-Szemerédi (RS) graphs have polynomially equivalent extremal densities for linear-sized matchings. Concretely, if ORS(n, εn) ≥ Ω(n^c), then for every fixed δ > 0, RS(n, Θ(ε^{1/δ} n)) ≥ Ω(n^{c(1−δ)}). The proof tensorizes a given ORS graph: for each k-tuple of matching indices, it forms a matching in the k-th tensor power, verifies via the suffix-induced ORS property that these matchings are induced, and applies a pigeonhole argument over the sum of indices. The author notes that this answers a question of Behnezhad and Ghafari and implies that the dynamic matching algorithm of Assadi–Khanna–Kiss can be stated in terms of ordinary RS graphs.","tokens_in":3907,"tokens_out":17644,"duration_ms":160621,"significance":"If correct, the result is a clean and useful equivalence: it rules out the possibility that ORS graphs are much denser than RS graphs, and it transfers the combinatorial bottleneck in recent fully dynamic matching algorithms to the classical RS quantity. The proof is self-contained, elementary, and short; the tensor construction and the coordinate-wise use of the ORS property are elegant. The note is fully rigorous apart from a harmless counting slip (see minor comments), and it resolves an explicitly posed question. The contribution is modest but appropriate for a note.","major_comments":[],"minor_comments":[{"comment":"The equality |M_a| = r^k is not correct for an undirected graph: each k-tuple of undirected edges from M_{a_1},...,M_{a_k} gives rise to 2^{k-1} unordered edges of H_s, so |M_a| = 2^{k-1} r^k. This does not invalidate the lemma because the matchings are larger than claimed; delete edges from each M_a to leave exactly r^k edges, which preserves the induced-matching property, and then the stated bound follows. Please correct the counting and include the deletion step.","section":"Section 2, Lemma 2.1"},{"comment":"The edge set E(H_s) is written with ordered pairs, which is inconsistent with the subsequent treatment of H_s as an undirected graph; define E(H_s) as the set of unordered pairs {U,V} satisfying the stated condition.","section":"Section 2, definition of H_s"},{"comment":"The abstract's quantities ORS(n) and RS(n) hide the constant ε in 'Ω(n)' matchings. Since the proof requires a fixed ε with ORS(n, εn) ≥ Ω(n^c), the abstract should either refer to the parameterized statement or explicitly state that the constant ε is fixed across n.","section":"Abstract and Theorem 1.3"},{"comment":"The phrase 'if any vertex U ∈ M_a' should be 'if any vertex U incident to an edge of M_a' for precision.","section":"Section 2, Lemma 2.1 proof"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a short note with an elementary proof. The only technical issue is the harmless factor-of-2^{k-1} miscount in Lemma 2.1; the theorem is unaffected. I see no obstacle to publication after the counting step is corrected. The result is likely of interest to the combinatorial algorithms community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kevin Pratt's note shows that Ordered Ruzsa-Szemerédi graphs cannot be much denser than ordinary Ruzsa-Szemerédi graphs: if ORS(n, εn) ≥ n^c, then RS(n, ε^{1/δ} n) ≥ n^{c(1-δ)} for any fixed δ. This answers the question Behnezhad and Ghafari left open, and it's a genuine contribution. The proof is a tensor product construction: from an ORS graph G, build a graph on k-tuples of vertices, keeping edges whose coordinate-wise matching indices sum to s. The partition into tuples with the same index vector gives matchings, and the ORS condition is used exactly to force inducedness. The counting via pigeonhole over s is clean, and the non-perfect-power handling is standard.\n\nThe paper is honest about borrowing the tensor trick from the matrix multiplication / cap set literature, and it cites the right sources. For a note, it's unusually self-contained: the relevant definitions are stated, and the proof is complete.\n\nSoft spots are minor. The edge set of H_s is written as a set of ordered pairs, which is a bit sloppy; if read literally it would double count edges, but the intended reading is unordered edges and then |M_a| = r^k is correct. Even on the literal reading you'd only get a constant factor, so the theorem is unaffected. I'd ask the author to clean up the notation in a revision, but that's not a mathematical issue.\n\nThe result matters mainly to people working on dynamic matching, since it lets them restate the AKK25 runtime as a function of RS density, but it should also interest anyone who studies RS graphs. The paper is short, correct, and answers an explicit open question. I'd send it to a serious referee and expect it to be accepted after minor revisions.","headline":"A short, clean proof that ORS and RS graphs have polynomially equivalent max density; the main argument is sound and the paper deserves refereeing.","tokens_in":4406,"tokens_out":7253,"would_cite":true,"duration_ms":65564,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C70","05C35","68R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that ordered Ruzsa-Szemerédi graphs are roughly as dense as ordinary Ruzsa-Szemerédi graphs, closing a gap that would have made the ordered notion strictly more powerful.","keywords":["ordered Ruzsa-Szemerédi graphs","Ruzsa-Szemerédi graphs","induced matchings","fully dynamic matching","approximate maximum matching","tensor power construction","combinatorial equivalence"],"falsifier":"A concrete counterexample would be a family of $n$-vertex ORS graphs with $\\mathrm{ORS}(n, \\varepsilon n) \\ge n^c$ for some fixed $c>0$ while $\\mathrm{RS}(n, \\Theta(\\varepsilon^{1/\\delta} n)) \\le n^{c(1-\\delta)-\\omega(1)}$ for some fixed $\\delta>0$; exhibiting such graphs would disprove the theorem's tightness. Alternatively, a direct computation of ORS and RS for the known constructions at growing $n$ that shows a superpolynomial gap would falsify the rough equivalence.","tokens_in":3541,"feed_emoji":"📐","tokens_out":4066,"duration_ms":36330,"temperature":0.7,"pith_summary":"The paper establishes that two graph families—ordered Ruzsa-Szemerédi (ORS) graphs and ordinary Ruzsa-Szemerédi (RS) graphs—have roughly the same maximum density when matchings are linear-sized. Specifically, if an $n$-vertex ORS graph can have $\\Omega(n^c)$ linear matchings, then some ordinary RS graph must have $\\Omega(n^{c(1-\\delta)})$ linear matchings for any fixed $\\delta>0$. This resolves a question of Behnezhad and Ghafari, who introduced ORS graphs in the context of fully dynamic approximate maximum matching algorithms. The equivalence matters because it lets the runtime of those algorithms be expressed in terms of the older, better-understood RS graphs, and it rules out the possibility that the ordered relaxation is dramatically denser.","feed_headline":"Ordered Ruzsa-Szemerédi graphs match ordinary ones in density","feed_subtitle":"The ordered variant used in dynamic matching algorithms cannot be much denser than the classic construction.","key_machinery":"The construction $H_s$ is the central object. Given an ORS graph $G = M_1 \\cup \\cdots \\cup M_t$ on $n$ vertices, $H_s$ has vertex set $V(G)^k$, and an edge is a pair of $k$-tuples $(U,V)$ such that each coordinate $(U_i,V_i)$ is an edge of $G$ and the matching indices $f(U_i,V_i)$ sum to $s$. The edges partition into matchings $M_a$ indexed by $k$-tuples $a$ with $\\sum a_i = s$, and each $M_a$ is shown to be an induced matching by the coordinate-wise inequality $f(U_i,V_i) \\le a_i$ that follows from the ORS suffix condition. Pigeonhole averaging over $s$ yields many matchings.","core_discovery":"The central claim is Theorem 1.3: if $\\mathrm{ORS}(n, \\varepsilon n) \\ge \\Omega(n^c)$, then for every fixed $\\delta > 0$, $\\mathrm{RS}(n, \\Theta(\\varepsilon^{1/\\delta} n)) \\ge \\Omega(n^{c(1-\\delta)})$. This is proved by a tensor-power construction: from an $n$-vertex ORS graph with $t$ matchings of size $r$, one builds an $n^k$-vertex graph whose edges are $k$-tuples of ORS edges whose matching indices sum to a fixed value $s$. That graph is partitioned into $t^{k-1}/k$ matchings of size $r^k$, each of which is an induced matching in the full graph. The proof exploits the ORS 'suffix-induced' property to bound each coordinate of an edge in the induced subgraph, then uses equality of the coordinate sums to force exact membership. As a consequence, the maximum density of ORS graphs with linear matchings is roughly equivalent to the same quantity for ordinary RS graphs.","pith_inferences":["The tensor-power loss of $\\delta$ in the exponent is likely inherent to the proof technique, but the same construction may transfer other 'suffix-induced' graph properties to ordinary induced matchings, potentially useful for other dynamic data structure problems.","One could conjecture a sharper equivalence, namely $\\mathrm{ORS}(n, \\varepsilon n) = \\mathrm{RS}(n, \\Theta(\\varepsilon n))^{1-o(1)}$, meaning the two quantities are subpolynomially equivalent even without fixing a polynomial regime.","A practical testable extension would be to compute the best known ORS and RS constructions side by side for moderate $n$ to see whether the ratio grows slowly or remains near 1, which would hint at whether the polynomial loss is an artifact of the proof."],"forward_implications":["If $\\mathrm{ORS}(n, \\varepsilon n) \\ge \\Omega(n^c)$, then $\\mathrm{RS}(n, \\Theta(\\varepsilon^{1/\\delta} n)) \\ge \\Omega(n^{c(1-\\delta)})$ for any fixed $\\delta > 0$.","The fully dynamic matching algorithm of Assadi, Khanna, and Kiss can be restated with update time $n^{o(1)} \\cdot \\mathrm{RS}(n, \\Theta_\\varepsilon(n))$, tying dynamic matching performance directly to the classic RS quantity.","It is impossible that $\\mathrm{RS}(n) = n^{o(1)}$ while $\\mathrm{ORS}(n) = n^{1-o(1)}$; the two notions must have the same asymptotic behavior up to arbitrarily small polynomial factors.","If the dynamic matching algorithm's update time turns out to be polynomial, then current constructions of both ORS and RS graphs are far from optimal."],"supporting_citations":[{"why":"Introduces ORS graphs and gives the dynamic matching algorithm whose runtime depends on $\\mathrm{ORS}(n, \\Theta_\\varepsilon(n))$.","marker":"[BG24]"},{"why":"Improves the dynamic matching algorithm to $n^{o(1)} \\cdot \\mathrm{ORS}(n, \\Theta_\\varepsilon(n))$, making the ORS-to-RS reduction directly relevant.","marker":"[AKK25]"},{"why":"Defines the classical Ruzsa-Szemerédi graphs, the notion that the paper shows is roughly equivalent to ORS.","marker":"[RS78]"},{"why":"Provides the lower bound $n^{o(1)} \\le \\mathrm{ORS}(n, \\varepsilon n)$, the starting point for understanding ORS density.","marker":"[FLN+02]"},{"why":"Gives the upper bound for ordinary RS graphs, establishing the known range that the new equivalence respects.","marker":"[Fox11]"},{"why":"Supplies the tensor-power trick that converts relaxed induced matchings into legitimate ones, which motivates Lemma 2.1.","marker":"[BCC+17]"},{"why":"Further develops the tensor-power construction that the proof adapts to the ORS setting.","marker":"[CFTZ22]"}],"fun_headline_variants":["Ordered Ruzsa-Szemerédi graphs match classic density","No density gap: ordered vs classic RS graphs","Ordered RS graphs as dense as classic","Density equivalence for ordered and classic RS graphs","Ordered RS graphs reach classic density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the ORS definition's suffix condition: each matching $M_i$ must be induced in the union of the matchings from $i$ onward, not merely in the whole graph; if the definition were weakened to whole-graph inducedness, the coordinate-wise bound $f(U_i,V_i) \\le a_i$ would not hold and the construction would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Ordered Ruzsa-Szemerédi graphs match classic density","No density gap: ordered vs classic RS graphs","Ordered RS graphs as dense as classic","Density equivalence for ordered and classic RS graphs","Ordered RS graphs reach classic density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001357,"raw_usage":{"total_tokens":5581,"prompt_tokens":1091,"completion_tokens":4490,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":4417}},"tokens_in":707,"tokens_out":4490,"duration_ms":34350,"temperature":1.0,"reasoning_tokens":4417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T12:06:04.534525+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be a family of $n$-vertex ORS graphs with $\\mathrm{ORS}(n, \\varepsilon n) \\ge n^c$ for some fixed $c>0$ while $\\mathrm{RS}(n, \\Theta(\\varepsilon^{1/\\delta} n)) \\le n^{c(1-\\delta)-\\omega(1)}$ for some fixed $\\delta>0$; exhibiting such graphs would disprove the theorem's tightness. Alternatively, a direct computation of ORS and RS for the known constructions at growing $n$ that shows a superpolynomial gap would falsify the rough equivalence.","supporting_citations":[],"review_version":1}