{"id":"48f17308-4a6c-4981-a5b7-f1d5a8c752c6","arxiv_id":"1908.08237","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For all graphs on at most four edges, the paper lists exact balance, strong-balance, and omnitonal numbers, and proves that the union of two bipartite graphs with the same edge count is balanceable.","lead":"This paper catalogues the balance, strong-balance, and omnitonal numbers for every graph with at most four edges, and proves that the disjoint union of two bipartite graphs with the same number of edges is always balanceable. It is a reference catalogue plus a structural theorem in Ramsey-type extremal graph theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 1's omnitonal entries rely on Theorem C with an unspecified n0(G); without an explicit bound, the advertised validity ranges are unquantified and the catalogue is incomplete.","rationale":"The reader's weakest assumption correctly identifies that Theorem C's unspecified n0 leaves the omnitonal table entries without quantified validity ranges. This is the most load-bearing concern because the paper's stated goal is to catalogue bal, sbal, and ot for all graphs on at most four edges; for the omnitonal entries, the paper does not deliver a computed threshold, only an existence statement. The concern does not invalidate Theorem 3.1—the proof of the union theorem is clean and self-contained—but it does mean the catalogue is incomplete as presented. The reader's CONDITIONAL verdict is appropriate: the main general result stands, while the catalogue requires either explicit n0 bounds or proofs of the amoeba property and the n0 values. I therefore agree with the reader's assessment and see no reason to shift the verdict. Other noted issues (e.g., omitted proofs for some bal entries, the inequality in Theorem 2.3 Case 2) are real but secondary; the n0 gap affects the widest class of entries and is the most central to the paper's deliverable.","tokens_in":13190,"tokens_out":40460,"duration_ms":348334,"concrete_test":"Take one representative entry, e.g., P5, and trace the proof of Theorem C in [6] to derive an explicit n0(P5). Verify that ot(n,P5)=ex(n,P5) for all n≥n0(P5) and compare n0(P5) with the smallest n for which a balanced copy can exist. Repeat for the other nine 'n≥n0' entries in Table 1. If any n0 cannot be bounded by an explicit function of |V(G)|, or if any derived n0 exceeds the ranges the table elsewhere relies upon, the table must be revised with stated bounds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central catalogue (Table 1) lists ot(n,G)=ex(n,G) for at least ten bipartite amoebas (4K2, 2K2∪K1,2, 2K1,2, K2∪P4, P5, K1,3 with extended leaf, P4, 3K2, P3∪K2, 2K2) with 'Valid n: n ≥ n0', citing Theorem C from [6]. Theorem C states only that some n0=n0(G) exists for bipartite amoebas; it gives no quantitative bound, and the paper never computes or estimates n0 for any of these graphs. The amoeba property itself is also asserted without proof for most of these graphs (only tK2 and Pk are mentioned in the text). Consequently, the table does not provide a usable validity range: a reader cannot tell for which n the equality holds, and if the true n0 exceeds any range the authors implicitly intend (e.g., the n≥10 or n≥12 used for other entries), those lines are effectively unverified. Since the omnitonal numbers are a major part of the paper's claimed catalogue, this missing quantification is a load-bearing gap rather than a cosmetic issue. The core union theorem (Theorem 3.1) appears sound, but the catalogue's completeness—a stated goal of the paper—is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies three Ramsey-type parameters for 2-edge-coloured complete graphs: the balance number bal(n,G), the strong balance number sbal(n,G), and the omnitonal number ot(n,G). Its two stated goals are to catalogue these parameters for all graphs with at most four edges (Tables 1–3) and to prove new general results. The main new theorem (Theorem 3.1) states that if G and H are bipartite graphs with e(G)=e(H), then the disjoint union G∪H is balanceable for n sufficiently large, with bal(n,G∪H) ≤ max{ex(n,G), ex(n,H)}. The paper also introduces a 'triple property' and uses it, together with Turán and Ramsey bounds, to prove that for n ≥ 7t−1, sbal(n,(2t−1)K2) = bal(n,2tK2) = bal(n,(2t+1)K2) = ex(n,tK2). Several ad-hoc theorems supply values for individual small graphs, and the paper closes with tables summarizing the computed values.","tokens_in":13480,"tokens_out":20629,"duration_ms":192485,"significance":"If fully supported, the catalogue of bal, sbal, and ot for all graphs on at most four edges would be a useful reference, and Theorem 3.1 is a genuine extension of earlier balanceability results: previous union theorems required one component to be balanceable or omnitonal, whereas Theorem 3.1 only requires bipartiteness and equal edge counts. The proof of Theorem 3.1 is short, elementary, and correct, and the matching formulas in Theorem 3.4 are a nice application. The paper is purely combinatorial, uses no fitted parameters, and, where it proves results directly, the arguments are mostly transparent. However, the advertised catalogue is not fully verified: several table entries rely on an existential n0 that is never quantified, on amoeba assertions that are not proved in the manuscript, and on case analyses that are left to the reader. These gaps affect the central claim that the paper provides a complete catalogue.","major_comments":[{"comment":"Table 1 lists ten omnitonal entries with 'Valid n: n ≥ n0' and cites Theorem C, but Theorem C only asserts the existence of n0 = n0(G) for bipartite amoebas; the paper never computes or bounds n0 for any of 4K2, 2K2∪K1,2, 2K1,2, K2∪P4, P5, K1,3 with extended leaf, P4, 3K2, P3∪K2, or 2K2. Since the stated goal is a catalogue with explicit validity ranges, a reader cannot determine for which n the equality ot(n,G)=ex(n,G) is claimed, and the advertised ranges are not established.","section":"Section 4, Table 1; Section 2.1, Theorem C"},{"comment":"The 'Amoeba' column in Table 1 marks 'Y' for the same ten graphs, but the amoeba property is not proved for them in the manuscript. The text explicitly mentions only tK2 and Pk as amoebas; Lemma G gives a necessary condition, not sufficiency, and the only reference for the amoeba status of the other graphs is [5], which is cited as 'in preparation'. Because Theorem C applies only to bipartite amoebas, the omnitonal entries remain unverified unless the amoeba property is supplied or a published reference is given.","section":"Section 4, Table 1; Section 2.1, Lemma G and ref. [5]"},{"comment":"In Theorem 2.2, Case 3, after constructing a K1,3 with red edges va, vb and blue edge vc, the proof says that 'all blue edges are incident with c with at most one more possible blue edge ab, making the total number of blue edges at most n'. This overlooks blue edges from v to Y = V(Kn)\\ {v,a,b,c}. Such edges do not immediately give the required vertex-disjoint blue K2, and they break the counting bound |B|≤n. The case analysis therefore does not establish ot(n,K1,3∪K2)=n as written.","section":"Theorem 2.2, Case 3"},{"comment":"In Theorem 2.3, Case 2, the condition for two independent red edges in the remaining graph is stated as (n−6 choose 2) − (n−5) ≥ n−6 and then simplified to n^2 −15n+52 ≥ 0. The correct equivalent inequality is n^2 −17n+64 ≥ 0, which fails for n=10 and n=11. Consequently the proof of bal(n,4K2)=n−1 does not justify the claimed range n≥10.","section":"Theorem 2.3, Case 2"},{"comment":"Footnote 1 to Table 2 states that the proofs for bal(n,2K1,2)=1, bal(n,K2∪P4)=1, bal(n,P5)=1, bal(n,K1,3 with extended leaf)=1, and bal(n,K3+e)=1 are 'very similar to the proof of Theorem 2.4' and are left to the reader. For a paper whose stated contribution is a complete catalogue, these entries are part of the central claim; deferring their proofs makes the catalogue incomplete. They should either be proved in the manuscript or the table should be presented as conjectural.","section":"Section 4, Table 2, footnote 1"}],"minor_comments":[{"comment":"The expression '|X| = n5 ≥ 5' appears to be a typo for 'n−5'.","section":"Theorem 2.2, Case 2(ii)"},{"comment":"The definition of the triple property uses 'bal(F,n)' and 'bal( F, n )' inconsistently; the function bal is defined with the graph as the second argument.","section":"Section 3, triple-property definition"},{"comment":"The K1,2 constructions such as 'K1,2 on the vertices {c; d, e}' use the semicolon notation for the center, but in several places the subsequent colour arguments appear to require the other vertex as the center; please check and clarify the intended stars.","section":"Theorem 2.4"},{"comment":"The word 'atrting' in the last sentence of the proof is a typo for 'starting'.","section":"Theorem 2.7"},{"comment":"In the proof of the triple property for matchings, the statement that the graph L on at least 3t+4 vertices is all-red is true only after the preceding exclusion of blue edges incident with the e_i; the text should make that dependence explicit.","section":"Observation 3.3"}],"recommendation":"major_revision","confidential_remarks":"The central new theorem (Theorem 3.1) is sound, and the matching result in Theorem 3.4 is a valuable addition. The main weakness is the verification of the catalogue: a large part of the evidence rests on the same group's unpublished preprint [5] ('Amoebas, in preparation') and on [6], and the omnitonal entries depend on an unquantified n0. For a journal publication the authors should supply either complete proofs or a published reference for the amoeba assertions, and a quantitative or at least a clearly existential n0 for the omnitonal entries. This is a verifiability issue rather than a novelty concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one result worth knowing here is Theorem 3.1: if G and H are bipartite with equal edge counts, then G∪H is balanceable. The proof is a short Ramsey-plus-Turán argument and it is correct. That is genuinely new and gives a large class of balanceable graphs that were not covered by earlier criteria. The matching closed forms via the triple property (Theorem 3.4) are also a solid addition.\n\nThe rest of the paper is a catalogue of bal/sbal/ot for graphs with at most four edges. This is where the soft spots are. Several table entries are not actually verified in the paper. The omnitonal rows in Table 1 all say 'valid n ≥ n0' with n0 taken from Theorem C of [6]; the paper never bounds n0 for any of those graphs, so the reader is left with an existential threshold. For a catalogue, that is a genuine incompleteness. In addition, the amoeba property is asserted for most of those graphs with no proof or reference to a public source (the paper cites [5], 'in preparation'). Some case analyses are also sloppy: Theorem 2.2's lower bound claims a red 2-factor avoids a red K1,3∪K2, but that only rules out copies with three or four red edges; a (2,2) copy can still appear. The same theorem's Case 3 ignores blue edges from the center to Y. Theorem 2.3 has a counting error in Case 2 ('four such deleted edges incident to u' should be three). These gaps are almost certainly repairable, but as written the catalogue is not a complete, verified computation. The 'left to the reader' entries in Table 2 are a further sign of that.\n\nI want to be clear that the central new theorem holds up; the issues are in the tables and the supporting cases, not in the main idea. The citation pattern is fine—reliance on [6] is appropriate, though leaning on an 'in preparation' paper for the amoeba facts is a weakness.\n\nWho is this for? Someone working in zero-sum Ramsey or unavoidable patterns will want the union theorem and the matching closed forms. The catalogue is useful as a reference if the gaps are fixed, but it is not yet a trustworthy table.\n\nMy recommendation: send it to peer review, but the referee should require the authors to either supply the missing n0 bounds (or explicitly state that no bound is known), justify the amoeba assertions, and repair the case analyses in 2.2 and 2.3 before publication. The paper has merit but needs a serious revision.","headline":"The union theorem is new and correct; the catalogue is not fully verified and needs repair before it can be trusted.","tokens_in":13975,"tokens_out":6897,"would_cite":true,"duration_ms":60778,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C35","05D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that equal-edge bipartite graph pairs always form a balanceable disjoint union, and completes all balance, strong-balance, and omnitonal values for graphs with at most four edges.","keywords":["balanceable graphs","strong balance number","omnitonal graphs","amoebas","edge colourings","Turán numbers","Ramsey numbers","matching graphs"],"falsifier":"A red/blue colouring of K_n in which each colour has more than max{ex(n,G),ex(n,H)} edges but no balanced copy of G∪H exists would refute Theorem 3.1. For the catalogue, computing the actual threshold n0 for a four-edge amoeba such as 4K2 and exhibiting a colouring with min{|R|,|B|} > ex(n,4K2) that lacks some required colour distribution would refute the corresponding table row.","tokens_in":12999,"feed_emoji":"⚖️","tokens_out":10214,"duration_ms":81644,"temperature":0.7,"pith_summary":"This paper works inside Ramsey theory: instead of asking when every red/blue edge-colouring of a large complete graph forces a monochromatic copy of a graph G, it asks when the colouring forces a copy of G whose colours are split evenly (or differ by one, when |E(G)| is odd). The main new result is that if G and H are any two bipartite graphs with the same number of edges, their disjoint union G∪H is balanceable, with bal(n,G∪H) ≤ max{ex(n,G),ex(n,H)} once n is at least |V(G)|+|V(H)|+R(G,H). This matters because individual bipartite graphs can fail to be balanceable—for instance C_{4t+2} is not—so the theorem produces balanceable unions out of non-balanceable parts. The paper also introduces a 'triple property' linking strong balance of (2t−1)K2 with balance of 2tK2 and (2t+1)K2, derives the exact common value $\\binom{t-1}{2} + (t-1)(n-t+1)$, and completes tables of bal, sbal, and ot for all graphs on at most four edges.","feed_headline":"Equal-edge bipartite pairs always balance together","feed_subtitle":"A new union theorem covers cases where no single bipartite graph balances—plus exact tables for graphs up to four edges.","key_machinery":"The argument turns on comparing colour-class sizes with extremal numbers. Theorem 3.1 uses the fact that for bipartite G and H, ex(n,G) and ex(n,H) are sub-quadratic, so a colour class with more than max{ex(n,G),ex(n,H)} edges must contain G (and the other must contain H); the Ramsey number R(G,H) then guarantees that the unclaimed vertex set can supply a monochromatic copy that makes the two copies vertex-disjoint. The 'triple property' is a second device: for G=(2t−1)K2, H=2tK2, F=(2t+1)K2, adding or deleting a single edge preserves the balance number, and the chain of inequalities reduces all three parameters to ex(n,tK2) via a classical extremal formula for matchings. The catalogue also leans on the amoeba concept—a graph that can be moved through a complete graph by single edge replacements—because Theorem C states that bipartite amoebas are omnitonal with ot(n,G)=ex(n,G).","core_discovery":"The central claim, Theorem 3.1, is that the disjoint union of two bipartite graphs with equal edge counts is balanceable. Concretely, if e(G)=e(H) and n ≥ |V(G)|+|V(H)|+R(G,H), then any 2-edge-colouring of K_n with more than max{ex(n,G),ex(n,H)} edges in each colour contains a balanced copy of G∪H. Because bipartite Turán numbers are sub-quadratic, the colour class with the larger count contains a copy of G and the other contains a copy of H; when the two copies overlap, the unused vertices still number at least R(G,H), so a monochromatic copy of one of the graphs can replace the overlapping part and separate the two copies. A second result, the triple property, shows that for G=(2t−1)K2, H=2tK2, F=(2t+1)K2 one has sbal(n,G)=bal(n,H)=bal(n,F) for all sufficiently large n, and Theorem 3.4 evaluates this common value as ex(n,tK2). These results, together with ad hoc arguments for specific four-edge graphs, yield the complete catalogue of bal, sbal, and ot for every graph with at most four edges.","pith_inferences":["Beyond the paper, the triple property suggests a general stability phenomenon: for many graphs G, adding one edge to a balanced graph family may leave bal unchanged, offering a route to exact balance numbers for larger graphs without fresh extremal analysis.","Beyond the paper, the union theorem opens a characterisation problem: given a family of bipartite graphs with equal edge counts, when is their disjoint union balanceable, and can bal(n, ∪G_i) be expressed in terms of the individual extremal numbers?","Beyond the paper, the unquantified n0 in Theorem C for four-edge amoebas invites a computational check: determining the first n where each listed graph becomes omnitonal would turn the table's asymptotic statements into explicit, machine-verifiable ranges."],"forward_implications":["Two copies of a non-balanceable bipartite graph, such as C_{4t+2}, form a balanceable union; the example in the paper gives bal(n, 2C_{4t+2}) ≤ (4t+1)n^{1+1/(4t+2)} + 16(4t+1)n for n ≥ 14t+6.","For matchings, the exact values for n ≥ 7t−1 are sbal(n,(2t−1)K2) = bal(n,2tK2) = bal(n,(2t+1)K2) = $\\binom{t-1}{2} + (t-1)(n-t+1)$.","Every graph on at most four edges now has a known balance number (when balanceable), with new values such as bal(n,4K2)=n−1 for n≥10 and bal(n,K1,3∪K2)=n−1 for n≥9.","The union theorem is a general construction: equal-edge bipartite pairs are balanceable without either component being balanceable, extending the class of graphs known to be balanceable."],"supporting_citations":[{"why":"Defines balanceable, strongly balanceable, and omnitonal graphs; supplies Theorems A–F, including Theorem C used for the omnitonal table rows.","marker":"[6]"},{"why":"Defines amoebas and contributes Lemma G on degree sequences, used to rule out omnitonal status for several four-edge graphs.","marker":"[5]"},{"why":"Classical formula for ex(n,tK2), used to evaluate the matching balance numbers in Theorem 3.4.","marker":"[11]"},{"why":"Known Ramsey number for t disjoint edges, used to set the threshold n ≥ 7t−1.","marker":"[10]"},{"why":"Survey establishing sub-quadratic extremal numbers for bipartite graphs, the engine of Theorem 3.1's existence step.","marker":"[13]"},{"why":"Cited with [6,14] for the fact that a colour class with more than the extremal number of edges contains a copy of the bipartite graph.","marker":"[4]"},{"why":"Turán-type theorem for unavoidable patterns, supporting the same existence step in Theorem 3.1.","marker":"[14]"},{"why":"Source of Lemma H on balanced type-A and type-B colourings, used to show C4 is not omnitonal and K3 is not strongly balanceable.","marker":"[7]"}],"fun_headline_variants":["Equal-edge bipartite unions always balance","Bipartite pairs with equal edges balance","Union of equal bipartite graphs balances","For equal-edge bipartite graphs, union balances"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The omnitonal rows of the catalogue depend on Theorem C's promise that ot(n,G)=ex(n,G) beyond some threshold n0(G), but the paper never computes or bounds any n0 for the listed four-edge amoebas, so the advertised 'n ≥ n0' validity ranges are unverified.","fun_headline_variants_meta":{"raw":{"variants":["Equal-edge bipartite unions always balance","Bipartite pairs with equal edges balance","Union of equal bipartite graphs balances","For equal-edge bipartite graphs, union balances"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000693,"raw_usage":{"total_tokens":3249,"prompt_tokens":1175,"completion_tokens":2074,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":791,"completion_tokens_details":{"reasoning_tokens":2020}},"tokens_in":791,"tokens_out":2074,"duration_ms":97898,"temperature":1.0,"reasoning_tokens":2020,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:47:44.481706+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A red/blue colouring of K_n in which each colour has more than max{ex(n,G),ex(n,H)} edges but no balanced copy of G∪H exists would refute Theorem 3.1. For the catalogue, computing the actual threshold n0 for a four-edge amoeba such as 4K2 and exhibiting a colouring with min{|R|,|B|} > ex(n,4K2) that lacks some required colour distribution would refute the corresponding table row.","supporting_citations":[{"cited_title":"Unavoidable chromatic patterns in 2-colorings of the complete graph","cited_arxiv_id":"1810.12375","evidence_quote":"Defines balanceable, strongly balanceable, and omnitonal graphs; supplies Theorems A–F, including Theorem C used for the omnitonal table rows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines amoebas and contributes Lemma G on degree sequences, used to rule out omnitonal status for several four-edge graphs."},{"cited_title":"Erd˝ os and T","cited_arxiv_id":null,"evidence_quote":"Classical formula for ex(n,tK2), used to evaluate the matching balance numbers in Theorem 3.4."},{"cited_title":"Cockayne and P.J","cited_arxiv_id":null,"evidence_quote":"Known Ramsey number for t disjoint edges, used to set the threshold n ≥ 7t−1."},{"cited_title":"F¨ uredi and M","cited_arxiv_id":null,"evidence_quote":"Survey establishing sub-quadratic extremal numbers for bipartite graphs, the engine of Theorem 3.1's existence step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of Lemma H on balanced type-A and type-B colourings, used to show C4 is not omnitonal and K3 is not strongly balanceable."}],"review_version":1}