{"id":"86867b39-6d6b-441a-aef2-f0769f8a99f1","arxiv_id":"2606.06258","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves Erdős-Hajnal conjecture for E-graph and Bird forbidden induced subgraphs via generalized iterative sparsification and reductions to generalized nice and (*) properties.","lead":"This paper proves the Erdős-Hajnal conjecture holds for graphs with no induced E-graph or Bird graph. A smart generalist might read it to track incremental progress on a major open problem linking forbidden induced subgraphs to guaranteed large cliques or independent sets.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the reduction and technical condition as central, but the full-text outline provides no concrete counterexample or inconsistency in those steps. The approach generalizes prior results on P5 and bull in a manner consistent with the field, so no adjustment to UNVERDICTED is warranted.","tokens_in":1894,"tokens_out":235,"duration_ms":16252,"concrete_test":"Re-derive the reduction from EH to generalized nice property (as described in the proof outline) and check whether the technical condition for E-graph and Bird follows directly from the cited Ramsey application and embedding construction; if the derivation holds without additional unstated assumptions, the claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The outlined strategy reduces EH to the generalized nice property (up to a technical condition addressed via Ramsey and a new embedding for leaf-free graphs), then to property (*), with an auxiliary graph shown to satisfy EH via equivalence relations. This follows the established iterative sparsification framework without apparent internal gaps or unsecured steps in the high-level argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to prove the Erdős-Hajnal conjecture for graphs with no induced E-graph (five-vertex path plus a pendant edge on the middle vertex) and no induced Bird graph (bull plus a pendant edge on one horn). It generalizes the iterative sparsification framework of Nguyen-Scott-Seymour by reducing (up to a technical condition) to a generalized nice property, verified for the target graphs via the Ramsey theorem and a new embedding for leaf-free graphs; it then reduces the generalized nice property to property (*) and verifies the latter for E-graph and Bird via equivalence relations on auxiliary graphs.","tokens_in":1954,"tokens_out":386,"duration_ms":21921,"significance":"If correct, the result adds two new forbidden induced subgraphs to the list of cases where the Erdős-Hajnal conjecture is known to hold, directly extending the five-vertex path result of Nguyen-Scott-Seymour and the bull result of Chudnovsky-Safra. The generalization of the nice property and the embedding technique for graphs without leaves are methodological contributions that may apply more broadly. The argument follows the established framework without introducing free parameters or ad-hoc axioms.","major_comments":[],"minor_comments":[{"comment":"Abstract: correct the typographical errors 'generlaized' (should be 'generalized'), 'ues' (should be 'use'), and 'satisfiy' (should be 'satisfy').","section":"Abstract"},{"comment":"Abstract (proof outline paragraph): the reduction steps and verification of the technical condition are described at a high level; add explicit cross-references to the sections or lemmas where each step (including the Ramsey application and the leaf-free embedding) is carried out.","section":"Abstract"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, recognition of the significance of the results, and recommendation of minor revision. No major comments are listed in the report, so we have no specific points requiring point-by-point rebuttal or revision at this stage. We will incorporate any minor editorial suggestions during the revision process.","responses":[],"tokens_in":1444,"tokens_out":81,"duration_ms":13845,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper establishes the Erdős-Hajnal conjecture for two new 6-vertex graphs: the E-graph (P5 plus a pendant edge on the middle vertex) and the Bird (bull plus a pendant on one horn). It generalizes the five-vertex path result and the bull result using an iterative sparsification approach.\n\nWhat works is the logical extension. They reduce EH (up to a technical condition) to a generalized nice property, prove the two graphs meet that condition via Ramsey theory plus a new embedding argument for leaf-free graphs, then reduce further to property (*) and verify it holds. They also handle an auxiliary graph through equivalence relations to confirm EH. The new graphs and the generalized property are genuine additions, and the high-level strategy follows the prior framework without obvious internal contradictions.\n\nThe soft spots are in the details that the abstract leaves out. The reductions and the new embedding step need line-by-line checking to confirm no hidden gaps in the technical condition or the equivalence relations. The soundness rating in the report is low precisely because the full verification is not visible from the summary, even though the stress-test finds no unsecured steps at the outline level. Typos in the abstract are minor but do not affect the math.\n\nThis is for specialists in induced-subgraph extremal problems who already know the Nguyen-Scott-Seymour and Chudnovsky-Safra papers. A reader tracking which graphs satisfy EH will find the two new cases useful. It deserves a serious referee because it adds concrete new instances and shows the framework can be stretched without breaking.","headline":"This extends EH to the E-graph and Bird by generalizing the nice property in the Nguyen-Scott-Seymour sparsification framework.","tokens_in":2419,"tokens_out":391,"would_cite":false,"duration_ms":12940,"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":"Graphs with no induced E-graph or Bird satisfy the Erdős-Hajnal conjecture.","keywords":["Erdős-Hajnal conjecture","induced subgraph","E-graph","Bird graph","forbidden induced subgraph","graph Ramsey theory","sparsification"],"falsifier":"A sequence of n-vertex graphs with neither an induced E-graph nor an induced Bird whose largest clique and largest independent set are both smaller than n to any fixed positive power.","tokens_in":2799,"feed_emoji":"","tokens_out":698,"duration_ms":18018,"temperature":0.7,"pith_summary":"The paper shows that the Erdős-Hajnal conjecture holds when the forbidden induced subgraph is either the E-graph or the Bird graph. The E-graph is formed by attaching a pendant edge to the middle vertex of a five-vertex path; the Bird is formed by attaching a pendant edge to one horn of a bull. This extends known cases for the plain five-vertex path and the bull. The argument reduces the conjecture, subject to a technical condition, to a generalized nice property, verifies the condition using Ramsey theory together with a new embedding method for leaf-free graphs, then reduces further to a property (*) that both graphs satisfy. One auxiliary step establishes the conjecture for certain auxiliary graphs by defining suitable equivalence relations.","feed_headline":"Erdős-Hajnal conjecture holds for E-graphs and Birds","feed_subtitle":"Two new forbidden induced subgraphs join the five-vertex path and bull as cases where every graph has a large clique or independent set.","key_machinery":"The generalized nice property, which extends the nice property from earlier work on the five-vertex path and serves as the intermediate target in the reduction of the Erdős-Hajnal conjecture.","core_discovery":"For the E-graph and the Bird graph, every n-vertex graph containing neither as an induced subgraph has a clique or independent set of size at least n^c for some positive c that depends only on the forbidden graph. The proof generalizes the iterative sparsification framework by first reducing (under a technical condition) the conjecture to the generalized nice property, confirming that E-graph and Bird meet the condition, then reducing the generalized nice property to a new property (*) and verifying that both graphs satisfy (*).","pith_inferences":["The same reduction chain might apply to other graphs obtained by adding pendant edges to paths or bulls.","If the generalized nice property can be shown for additional graphs, the conjecture would hold for those graphs as well.","The equivalence-relation technique for auxiliary graphs may simplify proofs for other small forbidden induced subgraphs."],"forward_implications":["The conjecture holds for all graphs forbidding the E-graph as an induced subgraph.","The conjecture holds for all graphs forbidding the Bird as an induced subgraph.","The iterative sparsification method extends from the five-vertex path to these two larger graphs.","Certain auxiliary graphs constructed during the proof also obey the Erdős-Hajnal conjecture.","The new embedding technique for graphs without leaves can be reused for other forbidden subgraphs."],"fun_headline_variants":["Erdős-Hajnal for E-graphs and Birds","E-graph and Bird satisfy Erdős-Hajnal","Extending Erdős-Hajnal to E-graph and Bird","E-graphs and Birds: Erdős-Hajnal cases"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the E-graph and Bird meet the technical condition allowing reduction of the conjecture to the generalized nice property.","fun_headline_variants_meta":{"raw":{"variants":["Erdős-Hajnal for E-graphs and Birds","E-graph and Bird satisfy Erdős-Hajnal","Extending Erdős-Hajnal to E-graph and Bird","E-graphs and Birds: Erdős-Hajnal cases"]},"model":"grok-4.3","cost_usd":0.005334,"raw_usage":{"total_tokens":2595,"prompt_tokens":869,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":53340500,"prompt_tokens_details":{"text_tokens":869,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1659,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":869,"tokens_out":67,"duration_ms":12632,"temperature":1.0,"reasoning_tokens":1659,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T00:24:02.035222+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A sequence of n-vertex graphs with neither an induced E-graph nor an induced Bird whose largest clique and largest independent set are both smaller than n to any fixed positive power.","supporting_citations":[],"review_version":1}