{"id":"906337dd-a94b-4eb8-a386-1993de041762","arxiv_id":"2608.11736","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Any finite Euclidean Ramsey set remains Ramsey after adjoining any point outside its affine hull.","lead":"Adjoining any single point outside the affine hull of a finite Euclidean Ramsey set preserves the Ramsey property, resolving a conjecture of Ivan, Leader, and Walters. The proof uses a cyclic product construction and Kříž's orbit-gluing theorem rather than the transitivity assumptions of earlier work.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is internally sound, with only the cited Kříž orbit-gluing theorem as an external dependency.","rationale":"The reader correctly identifies Kříž's orbit-gluing theorem as the sole external dependency, but I did not find a concrete place where the proof misapplies it or where an internal step fails. The construction of F_n, the verification of the equivalence-relation hypotheses, and the geometric calculation of S_n all check out. The proof is a valid reduction of Theorem 1.1 to Kříž's theorem plus standard Ramsey-product facts. Since the cited theorem is published and the application appears faithful, the central claim holds up under scrutiny. The only residual risk is that the quoted statement of Kříž's theorem might be stronger than the published result; however, the argument is robust even if only full-orbit gluing is available. Therefore the reader's ACCEPT verdict should stand unchanged.","tokens_in":4104,"tokens_out":33193,"duration_ms":370928,"concrete_test":"Check the original statement of Kříž [3, Thm. 4.1] in Proc. Amer. Math. Soc. 112 (1991): confirm it states partial-orbit gluing for arbitrary r with the hypotheses used here. If the theorem instead glues full b-orbits, verify that the proof goes through with r=n+1: the same F_n is U(E;z0,b,n+1)-Ramsey, and S_n remains contained in a single U-class, so the existing distance computation still yields the required Ramsey set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the proof of Theorem 3.2 step by step. Lemma 3.1 correctly extends a monochromatic copy of X to an E_C-embedding of C_n; the product theorem gives C_n^{n+1} as E_n^{n+1}-Ramsey; subset inheritance makes F_n E-Ramsey; the cyclic coordinate shift b is an isometry of F_n and respects the restricted equivalence E; and the distance computation for S_n is correct. The only step imported from outside the manuscript is Kříž's orbit-gluing theorem [3, Thm. 4.1], used to merge the E-classes of z0 and bz0. I found no misapplication: F_n is E-Ramsey, b is an isometry respecting E, and r=2 is applied to distinct points z0 and bz0. If the published theorem supports the quoted partial-orbit form, the argument goes through. Even if the theorem only supports gluing full b-orbits, replacing r=2 by r=n+1 would make the entire shifted orbit one U-class, so S_n would still be monochromatic; the distance computation is unchanged. Thus no internal contradiction or missing support was identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves Theorem 1.1: if X is a finite Euclidean Ramsey set and z is a point outside the affine hull of X, then X∪{z} is also Ramsey, thereby answering Conjecture 8 of Ivan, Leader, and Walters. The proof proceeds in two stages. Proposition 2.1 gives an elementary diagonal-product construction under the additional assumptions that the orthogonal projection of z onto aff(X) lies in conv(X) and that the height is sufficiently large. For the general case, Section 3 uses Kříž's E-Ramsey framework: Lemma 3.1 shows that a finite set C containing X is E_C-Ramsey, where X is the only nonsingleton class; a cyclic product construction then produces one-point extensions with arbitrarily small heights, and Kříž's orbit-gluing theorem is invoked to merge the color classes of two points, yielding a monochromatic copy of S_n. A final product step scales the height to any prescribed nonzero value. The argument is self-contained except for two cited theorems of Kříž and requires no transitivity assumption on X.","tokens_in":4281,"tokens_out":19432,"duration_ms":187118,"significance":"If correct, the paper resolves a natural open conjecture in Euclidean Ramsey theory and significantly extends the recent generalized-prism work of Ivan, Leader, and Walters. The proof is elegant: the elementary Proposition 2.1 is a nice standalone result, and the use of E-Ramsey configurations and orbit gluing in Section 3 is a technically clean way to remove the projection and height restrictions. All algebra and isometry computations in the manuscript check out, and the dependencies on external theorems are clearly stated. The paper also acknowledges independent work by Moore. I consider the result to be a solid contribution to the field.","major_comments":[],"minor_comments":[{"comment":"In the definition of ρ_X(y)^2, the quantity ρ_X(y) itself is never explicitly defined; the reader must infer from the inequality |λ| ≥ ρ_X(y) that ρ_X(y) is the square root of the right-hand side. Please make this explicit.","section":"Section 2, Proposition 2.1"},{"comment":"After applying the orbit-gluing theorem, the sentence 'The relation U(E;z0,b,2) merges the E-classes of z0 and bz0' is true because U contains E and the pair (z0,bz0); spelling this out would help readers unfamiliar with the construction.","section":"Section 3, proof of Theorem 3.2"},{"comment":"The note about the relationship to Moore's preprint is a bit unusual in the main text; it might be better placed in the acknowledgements or a footnote, although it is not problematic.","section":"General"},{"comment":"Please verify that the statement of Kříž's Theorem 4.1 as quoted in Section 3 indeed covers the partial-orbit gluing with arbitrary r≥1, since the proof applies it with r=2. If the theorem in [3] is only stated for full orbits, the proof still works by replacing r=2 with r=n+1; a brief remark to this effect would preempt any reader concern.","section":"Section 3, Kříž's theorem citation"}],"recommendation":"accept","confidential_remarks":"I recommend acceptance. The only point I would ask the authors to double-check is the exact formulation of Kříž's orbit-gluing theorem cited as [3, Theorem 4.1]; if it does not support the r=2 partial-orbit form used in the proof, the argument can be trivially repaired by setting r=n+1. This does not affect my positive assessment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper proves a conjecture of Ivan, Leader, and Walters, but Moore's preprint already proved the same theorem two days earlier. What's actually new here is the proof: a cyclic product construction combined with Kříž's orbit-gluing theorem. That technique is worth a look, and the paper is honest about the overlap.\n\nThe proof is complete and checks out. Lemma 3.1 correctly extends a monochromatic copy of X to an E_C-embedding of C_n, the product theorem gives C_n^{n+1} as E_n^{n+1}-Ramsey, and the subset inheritance step is valid. The cyclic shift b is indeed an isometry respecting the restricted equivalence, and the distance computation for S_n is correct. I verified the orbit-gluing application: F_n is E-Ramsey, b respects E, and r=2 merges the right classes. Even if only full orbits were supported, the argument still works with r=n+1. So the math is solid.\n\nThe soft spots are real but minor. The main theorem is not new: Moore's preprint [4] is cited and dated, and the paper acknowledges it. That means the incremental contribution is a new proof technique and confirmation of the conjecture, not the first proof. The paper also leans on two theorems of Kříž as black boxes, which is fine for a specialist paper but limits self-containedness. The remark about private circulation in June 2026 is a priority claim that can't be independently verified, but it's stated politely and doesn't affect correctness.\n\nWho is this for? Anyone working in Euclidean Ramsey theory or combinatorial geometry. The orbit-gluing trick is likely to be useful beyond this specific result. I'd bring it to a reading group and I'd send it to a serious referee. The overlap with Moore should be flagged to the editor, but a fresh proof of a conjecture with a genuinely new technique is publishable even if the theorem is known.\n\nRecommendation: accept for peer review, with the priority situation handled by the editors. This paper deserves a referee, not a desk reject.","headline":"New proof of a theorem Moore already proved, but the proof technique is original and the paper is honest about priority.","tokens_in":4797,"tokens_out":2073,"would_cite":true,"duration_ms":19756,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["05D10","52C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"One-point extensions of finite Euclidean Ramsey sets are Ramsey.","keywords":["Euclidean Ramsey sets","Ramsey theory","one-point extension","E-Ramsey configuration","orbit-gluing","cyclic product","affine hull"],"falsifier":"A counterexample would be a finite Ramsey set $X$, a point $z\\notin\\mathrm{aff}(X)$, and a colouring of every $\\mathbb{R}^N$ that avoids monochromatic copies of $X\\cup\\{z\\}$; the proof's cyclic construction is explicit enough that checking it on a regular-simplex base would reveal whether such a colouring can exist.","tokens_in":3897,"feed_emoji":"📐","tokens_out":10521,"duration_ms":106471,"temperature":0.7,"pith_summary":"A Euclidean Ramsey set is a finite configuration $X$ such that, for every $k$, some high-dimensional space has every $k$-colouring containing a monochromatic isometric copy of $X$. The paper proves that if $X$ is such a set and $z$ is any point outside the affine hull of $X$, then $X\\cup\\{z\\}$ is again Ramsey. This answers the conjecture posed in the recent study of generalized prisms, which had only been proved under a stronger assumption on $X$. The consequence is a closure property: finitely many points can be attached one at a time outside the current affine span without ever leaving the class of Ramsey sets. Notably, the proof uses no transitivity assumption on the base $X$.","feed_headline":"Any one-point extension of a Ramsey set is Ramsey","feed_subtitle":"The proof closes a conjecture about prisms, with no transitivity assumption on the base.","key_machinery":"The load-bearing construction is the diagonal embedding $D(x)=(x,a_1,\\dots,a_n)$ inside the $(n+1)$-fold product $C_n^{n+1}$, where $a_i=(1-i/n)x_0+(i/n)y$ are equally spaced points on the segment from $x_0$ to $y$. A cyclic coordinate shift $b$ acts on the product and respects the equivalence relation whose only nontrivial class is $X$. The orbit-gluing theorem then forces the class of $D(X)$ and the class of $bz_0$ to be monochromatic in every sufficiently high-dimensional colouring. The identity $\\|D(x)-bz_0\\|^2=\\|x-y\\|^2+\\|y-x_0\\|^2/n$ shows the added height is $\\|y-x_0\\|/\\sqrt{n}$, which can be made arbitrarily small; the product theorem then scales this configuration up to any prescribed nonzero height.","core_discovery":"The central claim is Theorem 1.1: for a finite Ramsey set $X\\subseteq\\mathbb{R}^d$ and any point $z\\notin\\mathrm{aff}(X)$, the extended set $X\\cup\\{z\\}$ is Ramsey. The proof splits into two tiers. When the orthogonal projection of $z$ lies in $\\mathrm{conv}(X)$ and its height is at least the weighted radius $\\rho_X(y)$, a direct diagonal product yields a monochromatic copy. The general case is handled with the language of $E$-Ramsey configurations, where an equivalence relation $E$ tells which points must share a colour. The construction places $n+1$ equally spaced points from an $x_0\\in X$ to $y$, forms an $(n+1)$-fold product, uses a cyclic shift to glue the diagonal copy of $X$ to the shifted apex through the orbit-gluing theorem, and obtains an isometric copy of $X\\cup\\{z\\}$. The height produced by the cyclic step is $\\|y-x_0\\|/\\sqrt{n}$, which shrinks to $0$; multiplying by a two-point set then reaches any prescribed nonzero height.","pith_inferences":["The orbit-gluing template may apply beyond one-point extensions, for instance to attach several exterior points at once when the base has additional symmetries; the paper does not pursue this.","The fact that heights tend to zero suggests the Ramsey property of such extensions is scale-invariant in the direction of the apex; whether arbitrarily small heights already suffice for every finite Ramsey base is a natural open question.","One could test the construction computationally on small bases such as regular simplices: the equivalence classes in the product are explicit, so a direct search could reveal whether the orbit-gluing step gives tight dimension bounds."],"forward_implications":["Every pyramid with a finite Ramsey base is Ramsey: the apex may be any point outside the base's affine hull.","The Ramsey class is closed under sequential one-point extensions, so any finite set grown from a Ramsey base by adding points outside the current affine span is again Ramsey.","The proof needs no transitivity or subsolubility of the base, removing the main restriction from the earlier generalized-prism construction.","Because the construction realizes arbitrarily small apex heights and then rescales, the size of the height is never an obstruction.","Together with the known closure of Ramsey sets under products and scaling, this yields many new explicit Ramsey sets in high dimensions."],"supporting_citations":[{"why":"Supplies the standard closure facts (scaling invariance, subset inheritance, finite products) used throughout the proof.","marker":"[1]"},{"why":"Formulates the conjecture and the generalized-prism result whose transitivity assumption this paper removes.","marker":"[2]"},{"why":"Supplies the E-Ramsey product theorem and the orbit-gluing theorem that power the general case.","marker":"[3]"}],"fun_headline_variants":["Adding any outside point keeps a Ramsey set Ramsey","Conjecture solved: one-point extensions stay Ramsey","Any point beyond affine hull extends a Ramsey set","Ramsey sets closed under one-point extensions","No transitivity needed for Ramsey point prisms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's general case imports the orbit-gluing theorem as a black box, and the entire argument collapses if that theorem does not apply to the cyclic shift and equivalence relation used here.","fun_headline_variants_meta":{"raw":{"variants":["Adding any outside point keeps a Ramsey set Ramsey","Conjecture solved: one-point extensions stay Ramsey","Any point beyond affine hull extends a Ramsey set","Ramsey sets closed under one-point extensions","No transitivity needed for Ramsey point prisms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1230,"prompt_tokens":878,"completion_tokens":352,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":281}},"tokens_in":494,"tokens_out":352,"duration_ms":3982,"temperature":1.0,"reasoning_tokens":281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:29:21.522785+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A counterexample would be a finite Ramsey set $X$, a point $z\\notin\\mathrm{aff}(X)$, and a colouring of every $\\mathbb{R}^N$ that avoids monochromatic copies of $X\\cup\\{z\\}$; the proof's cyclic construction is explicit enough that checking it on a regular-simplex base would reveal whether such a colouring can exist.","supporting_citations":[{"cited_title":"Erd˝ os, R","cited_arxiv_id":null,"evidence_quote":"Supplies the standard closure facts (scaling invariance, subset inheritance, finite products) used throughout the proof."},{"cited_title":"Generalised Prisms and Euclidean Ramsey Theory","cited_arxiv_id":"2606.13472","evidence_quote":"Formulates the conjecture and the generalized-prism result whose transitivity assumption this paper removes."},{"cited_title":"Kˇ r´ ıˇ z,Permutation groups in Euclidean Ramsey theory, Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the E-Ramsey product theorem and the orbit-gluing theorem that power the general case."}],"review_version":1}