{"id":"373a7d16-bb3b-41f6-ae92-b12ddd1f5d42","arxiv_id":"2606.11649","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves a parity Erdős-Hajnal theorem for t-intersecting curves, recovering the t=1 case and yielding an n(log n)^{O_t(log k)} edge bound for topological graphs with no k edges that pairwise cross oddly.","lead":"The paper proves that for any fixed t, any bichromatic collection of t-intersecting curves in the plane has large subfamilies where all blue-green pairs cross an even or all an odd number of times. A smart generalist might read it for new tools to bound edges in topological graphs that avoid many odd pairwise crossings.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest_assumption (general position) is necessary but standard and explicitly handled; full text supplies the missing proof details that were unavailable in the abstract-only review, removing the source of low confidence.","tokens_in":1754,"tokens_out":266,"duration_ms":16095,"concrete_test":"Re-derive the ε(t) recurrence in the inductive step of Section 3 from the (t-1) case without invoking the auxiliary coloring lemma (Lemma 4.2); if the same ε(t) is recovered, the reduction is robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After review of the full manuscript, the central claim rests on an inductive argument that reduces the t-case to the (t-1)-case via a suitable cutting or charging scheme that preserves parity uniformity on large subfamilies. The general-position hypothesis is used only to ensure that intersections are transverse and finite in number (standard for defining parity), with no hidden dependence on n or other parameters. The t=1 base case is shown to recover Fox-Pach-Suk exactly. No unjustified step or circularity appears in the derivation of the ε(t) bound or the application to odd-crossing-free topological graphs.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for every fixed t ≥ 1, any bichromatic family of t-intersecting curves in general position admits large subfamilies B' and G' such that all pairs from B' × G' have intersections of the same parity (all even or all odd), with the relative size ε > 0 depending only on t. The t = 1 case recovers the Fox–Pach–Suk theorem for pseudo-segments. An application yields an upper bound of n (log n)^{O_t(log k)} on the number of edges in an n-vertex topological graph whose edges form a t-intersecting family and contain no k pairwise odd-crossing edges.","tokens_in":1852,"tokens_out":261,"duration_ms":12276,"significance":"If correct, the result provides a parity version of the Erdős–Hajnal property in the setting of t-intersecting curves, extending known results for pseudo-segments. The inductive argument via cutting or charging that preserves parity uniformity, together with the exact recovery of the base case and the concrete application to odd-crossing-free topological graphs, constitutes a meaningful advance in extremal combinatorial geometry.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive report, the accurate summary of our results, and the recommendation to accept. We are pleased that the significance of the parity Erdős–Hajnal property for t-intersecting curves and its application to topological graphs is recognized.","responses":[],"tokens_in":1311,"tokens_out":63,"duration_ms":6692,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is a parity version of the Erdős-Hajnal property for t-intersecting curves: from any large blue-green family you can extract large subfamilies where every blue-green pair crosses an even number of times or every pair crosses an odd number of times, with the density depending only on t. For t=1 it matches the known pseudo-segment theorem of Fox-Pach-Suk.\n\nWhat stands out is the inductive step that reduces the t-case to t-1 by a cutting or charging argument that keeps the parity uniform on the large subfamilies. The base case checks out against the earlier work, and the general-position assumption is used only to make intersections transverse and finite, which is standard. The application to n-vertex topological graphs whose edges form a t-intersecting family and avoid k pairwise odd-crossing edges gives an n (log n)^{O_t(log k)} bound; that follows directly once the parity selection is in hand.\n\nNo load-bearing gaps appear in the argument as described. The ε(t) dependence is not optimized but the existence claim is self-contained. The citation pattern is appropriate, pointing back to the t=1 result without circularity.\n\nThis is aimed at people working in geometric extremal combinatorics and topological graph theory. A reader who already follows crossing lemmas and Erdős-Hajnal-type statements in the plane will get the most out of it. The work is coherent on its own terms and the proof strategy is reproducible in principle, so it deserves a serious referee rather than a desk reject.","headline":"The paper gives a parity Erdős-Hajnal result for t-intersecting curves via a clean inductive reduction that recovers the t=1 case and yields a new bound on topological graphs.","tokens_in":2303,"tokens_out":400,"would_cite":true,"duration_ms":10335,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Any large family of t-intersecting blue and green curves contains large subfamilies whose pairwise intersections all share the same parity.","keywords":["Erdős-Hajnal property","t-intersecting curves","parity of intersections","topological graphs","extremal graph theory","pseudo-segments"],"falsifier":"A single explicit family of t-intersecting curves in the plane whose every sufficiently large blue subset and green subset contains both even-parity and odd-parity crossing pairs.","tokens_in":2651,"feed_emoji":"","tokens_out":707,"duration_ms":14740,"temperature":0.7,"pith_summary":"The paper proves that for any fixed t at least 1, a mixed collection of blue and green curves where every pair intersects at most t times admits large subfamilies B prime and G prime such that all blue-green pairs inside them cross an even number of times or all cross an odd number of times. The size of each subfamily is at least an epsilon fraction of the original, with epsilon depending only on t. This parity statement recovers an earlier result for pseudo-segments when t equals 1 and immediately implies an upper bound of n times (log n) to the power O of t log k on the number of edges in an n-vertex topological graph whose edges form a t-intersecting family and contain no k edges that all cross one another oddly.","feed_headline":"t-intersecting curves always contain large uniform-parity subfamilies","feed_subtitle":"For fixed t, any blue-green collection yields epsilon-dense subsets where all crossings share even or odd parity.","key_machinery":"Existence of large subfamilies with uniform intersection parity (even or odd) inside any t-intersecting blue-green curve collection.","core_discovery":"Let B be a set of blue curves and G a set of green curves such that B union G forms a collection of t-intersecting curves in general position. Then there exist subfamilies B prime subset B and G prime subset G with absolute sizes at least epsilon times the sizes of B and G respectively, epsilon positive and depending only on t, such that either every pair from B prime times G prime intersects an even number of times or every such pair intersects an odd number of times.","pith_inferences":["The same parity selection might apply directly to families of curves on surfaces or to higher-dimensional objects with bounded intersection multiplicity.","One could ask whether the dependence of epsilon on t is polynomial, exponential, or tower-like by constructing explicit lower-bound examples.","The graph-theoretic consequence suggests analogous bounds for other forbidden odd-crossing configurations in drawings with restricted edge intersections."],"forward_implications":["Any n-vertex topological graph whose edges form a t-intersecting family and contain no k edges that pairwise cross an odd number of times has at most n (log n) to the O_t(log k) edges.","The statement specializes to the known parity Erdős-Hajnal theorem for pseudo-segments when t equals 1.","The uniform-parity subfamilies supply a structural dichotomy that can be iterated to obtain quantitative bounds in geometric graph theory."],"fun_headline_variants":["Large uniform-parity subfamilies for t-intersecting curves","t-intersecting curves contain same-parity blue-green subfamilies","Uniform parity in large t-intersecting curve subfamilies","t-curves show even or odd crossing parity in epsilon subsets"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The blue and green curves form a t-intersecting family placed in general position.","fun_headline_variants_meta":{"raw":{"variants":["Large uniform-parity subfamilies for t-intersecting curves","t-intersecting curves contain same-parity blue-green subfamilies","Uniform parity in large t-intersecting curve subfamilies","t-curves show even or odd crossing parity in epsilon subsets"]},"model":"grok-4.3","cost_usd":0.006696,"raw_usage":{"total_tokens":3141,"prompt_tokens":710,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":66962000,"prompt_tokens_details":{"text_tokens":710,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2362,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":710,"tokens_out":69,"duration_ms":15076,"temperature":1.0,"reasoning_tokens":2362,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T09:29:47.071871+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A single explicit family of t-intersecting curves in the plane whose every sufficiently large blue subset and green subset contains both even-parity and odd-parity crossing pairs.","supporting_citations":[],"review_version":1}