{"id":"44cd77d1-c4f6-4143-97dc-e5e5a65cab1e","arxiv_id":"1908.02348","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims Ramsey and Gallai-Ramsey numbers for stars with extra leaf edges, but the general proofs are unsound.","lead":"This paper studies two-color Ramsey and Gallai-Ramsey numbers for star-like graphs with extra edges between leaves. The small exact cases look promising, but the proofs of the general claims contain false steps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2(3) as stated contradicts the paper's own r=2 and r=3 results and is not the bound proved in §2.3; the Gallai-Ramsey Theorem 4 inherits an unsupported base case.","rationale":"The paper aims to determine Ramsey and Gallai-Ramsey numbers for stars plus independent edges. The general Ramsey theorem is the backbone, and Theorem 4's base case is exactly Theorem 2(3). The internal inconsistency between the statement (2t + 2r − 1) and the proof (2t − 1) is decisive: no repair within the text fixes Theorem 4 as stated. The reader's chosen weakest assumption (Dirac/Hamiltonicity) is plausible but should be relocated: in the proof as written on K_{2t−1}, a uniform red degree at least t would meet Dirac's threshold, so the Hamiltonicity step is not the false step; the false step is (a) the theorem/preamble mismatch and (b) the unjustified WLOG that the disjunctive degree condition yields a uniform color. We credit the detailed case analyses for r = 2, 3 and the t = 6, 8 Gallai-Ramsey computations, but those do not salvage the general theorem. The lower-bound construction for the stated 2t + 2r − 1 is absent; the construction given establishes only 2t − 1. Hence rejection remains appropriate.","tokens_in":28296,"tokens_out":13678,"duration_ms":132553,"concrete_test":"Specialize to r = 2, t = 7: evaluate Theorem 2(3), which predicts 17, against Theorem 2(1), which predicts 13. Since both are asserted for the same graph S^2_7, at most one can be correct; the coloring in §2.1 (two blue-joined K_6's) is already a 12-vertex coloring with no monochromatic S^2_7, so 13 is the better candidate. A complete verification would re-prove Theorem 2(3) with the corrected target 2t − 1 and check whether the §2.3 Case 2 subcase where some vertices are blue-high but not red-high is covered; if it is not, the upper bound is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.3 begins by saying it will prove R(S_t^r, S_t^r) = 2t − 1 and then works in K_{2t−1}, but Theorem 2(3) states the value 2t + 2r − 1. These cannot both be right: at r = 2, t ≥ 7, Theorem 2(1) gives 2t − 1 while Theorem 2(3) would give 2t + 3. The lower-bound construction in §2.1 uses two copies of K_{t−1} joined by blue edges, of order 2t − 2, which supports only the 2t − 1 value. Since Theorem 4's k = 2 case is the statement R(S_t^r, S_t^r) = 2t + 2r − 1, the general Gallai-Ramsey bounds rest on the false stated value. The reader's Dirac objection is a symptom: in the stated K_{2t+2r−1}, red minimum degree t does not imply a Hamilton cycle (e.g., a red K_{t, t+2r−1} has min degree t but is not Hamiltonian), and in the proof-as-written for K_{2t−1} the step from \"for every vertex |A_v| ≥ t or |B_v| ≥ t\" to \"WLOG |A_v| ≥ t for every v\" is unjustified. Either way the upper-bound proof does not establish the stated theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Ramsey and Gallai–Ramsey numbers for graphs S_t^r, obtained from a star K_{1,t-1} by adding r independent edges among leaves. It claims exact 2-color Ramsey numbers for r=2,3 and a general formula R(S_t^r,S_t^r)=2t+2r-1 for t≥6r-5, exact Gallai–Ramsey numbers for S_6^2 and S_8^2, and general upper and lower bounds for S_t^2 and S_t^r. The proofs use Gallai partitions, matching deletions, Dirac-type Hamiltonian arguments, and induction on the number of colors.","tokens_in":28586,"tokens_out":14782,"duration_ms":139851,"significance":"The family S_t^r interpolates between stars and fans, and exact values for small cases would be a useful addition to the Gallai–Ramsey literature. The paper's main structural idea—using the Gallai partition and bounding the number of large parts—is standard and potentially effective. However, the general base case in Theorem 2(3) is inconsistent with the paper's own r=2 theorem, and the lower bound for the stated value is absent. Because Theorem 4 inherits this base case, the paper's main general claims are not supported in its current form. No machine-checked proofs or reproducible code are provided; the contribution is entirely theoretical.","major_comments":[{"comment":"Theorem 2(3) states R(S_t^r,S_t^r)=2t+2r-1 for t≥6r-5, but this contradicts Theorem 2(1), which states R(S_t^2,S_t^2)=2t-1 for t≥7; since 6·2-5=7, both formulas apply at r=2 and differ. The proof in §2.3 also begins by saying the goal is to prove R(S_t^r,S_t^r)=2t-1 and works in K_{2t-1}, while the stated theorem concerns K_{2t+2r-1}. The lower-bound construction in §2.1, two copies of K_{t-1} joined in blue, has order 2t-2 and supports only R≥2t-1; no construction of order 2t+2r-2 is given. Thus the statement and proof of Theorem 2(3) are not about the same quantity, and the claimed value is unsupported.","section":"Theorem 2(3); §2.1; §2.3"},{"comment":"The proof of the general-r upper bound contains two unproved pivotal steps. In Case 1, after obtaining a red fan F_{2r} centered at x, the text states \"Since the degree of x is 2t-2, it follows that the red or blue degree of x is t-1+(2r+1)=t+2r\"; no derivation is given for this either/or degree conclusion. In Case 2, the inference from d_R(v)>n/2+r to \"every vertex v is contained in a red fan F_{2r}\" is asserted to follow by a combinatorial counting argument that is never supplied. Both steps are essential for producing the forbidden monochromatic S_t^r.","section":"§2.3, Case 1 and Case 2"},{"comment":"The base case k=2 for the exact Gallai–Ramsey value of S_6^2 is stated in Lemma 3 as \"precisely R(S_6^2,S_6^2)=15,\" but Theorem 2(1) only proves R(S_t^2,S_t^2)=2t-1 for t≥7. The value R(S_6^2,S_6^2)=15 is used without proof or reference, so the claimed exact value for gr_k(K_3;S_6^2) rests on an unestablished base case. The same gap affects Theorem 3(3) for t=6.","section":"Lemma 3; Theorem 3(1),(3)"},{"comment":"Theorem 4 is stated for t≥6r-5, and its proof in Lemma 8 uses \"From Item (3) of Theorem 2, R(S_t^r,S_t^r)=2t+2r-1\" as the k=2 base case. Since Theorem 2(3) is not established and is inconsistent with Theorem 2(1), the induction base for the general Gallai–Ramsey bounds in Theorem 4 fails. Consequently the upper bounds in Theorem 4 are unsupported even if the later partition arguments were correct.","section":"Theorem 4; Lemma 8"},{"comment":"The proof of Theorem 2(2) invokes R(F_3,F_3)=13 from the authors' own submitted paper [6]. With no published source or proof included, the r=3 result depends on an unavailable external result. If this is the only proof of that step, it needs a published reference or an independent argument within the paper.","section":"§2.2, Case 3"}],"minor_comments":[{"comment":"The notation is inconsistent: S_t^+ and S^t appear in places that should read S_t^2 (e.g., A.1, Subcase 1.1), and R(S_8^2)=15 should be R(S_8^2,S_8^2)=15.","section":"Throughout Appendix A"},{"comment":"In the q=0 paragraph, the text reads \"R(S_6^2,S_6^2)=11\"; this should presumably be 15, and as written it is inconsistent with the base case used elsewhere in the same lemma.","section":"Lemma 3, k=3 case"},{"comment":"The displayed triangles \"uvu1v, u1u2w1u1 and u1u4w2u1\" are garbled; the intended triangles should be written with standard vertex notation so the contradiction is readable.","section":"§2.2, Claim 2"},{"comment":"The parameter s appears in expressions such as n(k-2,s,t) in several places and should be r.","section":"Lemma 8"},{"comment":"The introduction states that proofs of Item (3) of Theorem 3 and Theorem 4 are omitted, but they are included in Appendix A; the text should be updated to reflect the actual structure.","section":"Introduction and Appendix"}],"recommendation":"reject","confidential_remarks":"The manuscript contains several internal contradictions, most seriously the discrepancy between Theorem 2(1) and Theorem 2(3) and the unsupported base case for Theorem 4. These are not local typographical issues; the main general claims are not established in the present form. The paper may be worth resubmitting after a substantial reworking of Section 2.3 and of the supplied base cases, but in its current form it should not be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThe headline: the paper's general theorem is not supported and is in fact inconsistent with its own small-case results. Theorem 2(3) states R(S_t^r,S_t^r)=2t+2r−1, but Theorem 2(1) and (2) give 2t−1 for r=2,3; plugging r=2 into (3) gives 2t+3. The proof in §2.3 is doubly confused: it opens saying it will prove 2t−1, then derives a 'Claim 4' for 2t+2r−1; the lower-bound construction used throughout gives only 2t−2 vertices (so R≥2t−1), not 2t+2r−2. The step from 'each vertex has red or blue degree ≥ t' to 'WLOG every vertex has red degree ≥ t' is simply false, and the Dirac/Hamilton/fan-counting argument in Case 2 does not justify the conclusion. Theorem 4 inherits all of this because its k=2 base is Theorem 2(3).\n\nWhat is worth keeping: the family S_t^r is a natural interpolation between stars and fans, and the exact Gallai-Ramsey values for S_6^2 and S_8^2 are new. The lower-bound constructions are the usual blow-up of the 2-colored K5, but the exact order counts are non-trivial. If the small cases survive a careful check, they are real contributions to the catalog.\n\nThe soft spots beyond the general theorem: Lemma 3 uses R(S_6^2,S_6^2)=15 as the k=2 base, but Theorem 2(1) only covers t≥7 and the stated formula in Theorem 3(1) evaluates to 11 at k=2; the lower-bound construction in Lemma 2 has order 10, not 11, so the base case is unproved or the formula is wrong. The r=3 proof leans on their own submitted fan paper for R(F3,F3)=13; not circular, but not self-contained. The manuscript says proofs of Theorem 3(3) and Theorem 4 are omitted, yet they appear in an appendix; that is fine, though the appendix has typos and hand-wavy steps, e.g., 'using a combinatorial counting argument' in Claim 4.\n\nWho it's for: readers who want the S_6^2 and S_8^2 numbers may find the computations useful, but anyone relying on Theorem 4 for general r should not. My recommendation is to send it to referees — the small-case material deserves scrutiny and the general claims need either a corrected statement or removal. If I were handling it, I'd ask for a major revision that fixes the inconsistency, supplies the missing lower bound, and tightens the sloppy steps.","headline":"The general theorem contradicts the paper's own r=2,3 results and is not proved; the small-case Gallai-Ramsey computations are new but need a careful check.","tokens_in":29136,"tokens_out":8975,"would_cite":false,"duration_ms":79575,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $t\\ge 6r-5$, every red-blue coloring of $K_{2t+2r-1}$ contains a monochromatic copy of $S_t^r$, a star of order $t$ with $r$ extra independent leaf edges, and this threshold is sharp.","keywords":["Ramsey number","Gallai-Ramsey number","star with extra independent edges","edge coloring","rainbow triangle","Gallai partition","monochromatic subgraph","fan graph"],"falsifier":"Search for a red-blue coloring of $K_{2t+2r-1}$ with no monochromatic $S_t^r$ at the first value $t=6r-5$; finding one refutes the theorem. In the absence of an exhaustive search, check whether the red graph $K_{t,t+2r-1}$ (minimum red degree $t$, no spanning cycle) can be completed to such a coloring, since the printed upper-bound proof relies on that spanning cycle.","tokens_in":28075,"feed_emoji":"⭐","tokens_out":16128,"duration_ms":150202,"temperature":0.7,"pith_summary":"This paper tries to determine how many vertices force a monochromatic copy of $S_t^r$, the graph obtained from a star of order $t$ by adding $r$ independent edges between its leaves. It proves exact 2-color Ramsey numbers for $S_t^r$ in three regimes, the largest being $R(S_t^r,S_t^r)=2t+2r-1$ whenever $t\\ge 6r-5$, and it gives exact Gallai-Ramsey formulas for two small cases plus general upper and lower bounds for all $r$. Because $S_t^r$ sits between ordinary stars ($r=0$) and fans ($r=(t-1)/2$), these results connect two well-studied families and show that the extra leaf edges change the forcing threshold only linearly in $r$ while the Gallai-Ramsey numbers keep their characteristic $5^{k/2}$ growth. The paper's main message is that the decorated-star problem is governed by the same Gallai-partition recursion that governs stars and fans.","feed_headline":"Exact two-color threshold for a decorated star: 2t+2r-1","feed_subtitle":"The Gallai-Ramsey versions grow like 5^{k/2}; exact values settle t=6 and t=8 for r=2.","key_machinery":"The engine is the Gallai partition theorem: in any edge-coloring of a complete graph with no rainbow triangle, the vertices split into parts with at most two colors between parts, so the reduced graph is a 2-colored complete graph. The classical Ramsey proofs work vertex-by-vertex, splitting the neighborhood of a vertex into red and blue sets and forcing a monochromatic $S_t^r$ out of matchings, triangles, and fans inside those sets. The Gallai-Ramsey upper bounds iterate a base-5 blow-up construction: copies of a smaller color-free graph are arranged as the five parts of the unique 2-coloring of $K_5$ with no monochromatic triangle, which is what produces the $5^{k/2}$ factors in the bounds.","core_discovery":"The central discovery is that for $t\\ge 6r-5$ the 2-color Ramsey number of $S_t^r$ is exactly $2t+2r-1$: every red-blue coloring of $K_{2t+2r-1}$ contains a monochromatic $S_t^r$, while some coloring of $K_{2t+2r-2}$ avoids it. For $r=2$ and $r=3$ the paper obtains the smaller value $2t-1$ when $t\\ge7$ and $t\\ge15$, respectively. On the Gallai-Ramsey side, the paper proves lower bounds of order $2(t-1)5^{k/2}$ for even $k$ and $(t-1)5^{(k-1)/2}$ for odd $k$, with upper bounds differing only by constants linear in $r$, and in the cases $S^2_6$ and $S^2_8$ it derives exact formulas for every $k$.","pith_inferences":["The printed lower-bound argument for the general case reuses the $2t-2$-vertex two-clique example, which is too small to support the stated $2t+2r-1$ threshold; supplying a genuine $2t+2r-2$-vertex construction is the most direct way to make the theorem self-contained.","The spanning-cycle step used in the general upper-bound proof needs more red neighbors than the proof assumes; if that degree condition is tightened to about $t+r-1/2$, the same counting argument likely goes through and the theorem would survive in essentially the same form.","Because every lower-bound construction uses the unique 2-colored $K_5$ with no monochromatic triangle, any graph in this family shares the $5^{k/2}$ growth; a natural testable extension is to seek exact Gallai-Ramsey formulas for $S_t^r$ with $r\\ge3$ and small $t$.","A computational Ramsey search targeted at the first case $t=6r-5$ could independently check the claimed equality before the proof gap is repaired."],"forward_implications":["For fixed $r$ and all $t\\ge 6r-5$, the 2-color threshold is exactly $2t+2r-1$, so adding $r$ independent leaf edges raises the star Ramsey number by an additive term linear in $r$.","For $r=2$ and $r=3$, the threshold is $2t-1$, meaning these extra edges do not raise the star Ramsey number in those regimes.","The Gallai-Ramsey lower and upper bounds are within additive terms linear in $t$ and $r$, so the exponential-in-$k$ growth rate is $5^{k/2}$.","The exact formulas for $S^2_6$ and $S^2_8$ give the first complete Gallai-Ramsey answers for non-star, non-fan members of this family."],"supporting_citations":[{"why":"Supplies the Gallai-partition theorem: every rainbow-triangle-free coloring of a complete graph has a partition with at most two colors between parts, the structural base of all the Gallai-Ramsey upper bounds.","marker":"[1, 4, 5]"},{"why":"Restates the Gallai-partition structure in graph-coloring language and provides the reduced-graph viewpoint used to turn Gallai-Ramsey problems into iterated two-color Ramsey problems.","marker":"[5]"},{"why":"Provides the fan Gallai-Ramsey results and the value $R(F_3,F_3)=13$ plus structural facts that the paper adapts for the star-with-edges graphs.","marker":"[6]"}],"fun_headline_variants":["Decorated star Ramsey number pinned: 2t+2r-1","Exact Ramsey threshold for stars with added leaves","Gallai-Ramsey stars: bounds tight, exact for t=6,8","S_t^r Ramsey formula: 2t+2r-1 when t≥6r-5"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The general-$r$ proof depends on a spanning-cycle step that needs more red neighbors than the proof assumes, and on a lower-bound example that is smaller than the stated threshold.","fun_headline_variants_meta":{"raw":{"variants":["Decorated star Ramsey number pinned: 2t+2r-1","Exact Ramsey threshold for stars with added leaves","Gallai-Ramsey stars: bounds tight, exact for t=6,8","S_t^r Ramsey formula: 2t+2r-1 when t≥6r-5"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000401,"raw_usage":{"total_tokens":2072,"prompt_tokens":906,"completion_tokens":1166,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1080}},"tokens_in":522,"tokens_out":1166,"duration_ms":58160,"temperature":1.0,"reasoning_tokens":1080,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:50:02.814645+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a red-blue coloring of $K_{2t+2r-1}$ with no monochromatic $S_t^r$ at the first value $t=6r-5$; finding one refutes the theorem. In the absence of an exhaustive search, check whether the red graph $K_{t,t+2r-1}$ (minimum red degree $t$, no spanning cycle) can be completed to such a coloring, since the printed upper-bound proof relies on that spanning cycle.","supporting_citations":[{"cited_title":"Gy´ arf´ as and G","cited_arxiv_id":null,"evidence_quote":"Restates the Gallai-partition structure in graph-coloring language and provides the reduced-graph viewpoint used to turn Gallai-Ramsey problems into iterated two-color Ramsey problems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fan Gallai-Ramsey results and the value $R(F_3,F_3)=13$ plus structural facts that the paper adapts for the star-with-edges graphs."}],"review_version":1}