{"id":"aedde37f-c7f9-4d5a-b8a9-e1a2277ee5a7","arxiv_id":"2502.00135","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In the hypercube, the rainbow extremal number of any k-edge tree is conjectured to depend only on k, and the paper verifies this for several infinite families.","lead":"This paper proposes a rainbow analogue of the Erdős-Sós conjecture inside the hypercube: the largest properly edge-colored subgraph without a rainbow copy of a k-edge tree should depend only on k. It proves this for paths, trees with many leaves, and two families of spiders, and gives a general upper bound for all trees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conjecture 2 is false as stated when n < k-1; e.g. n=2, T=P_4 (a 4-edge path) has ex*(Q_2,P_4) ≤ 4 but the conjecture predicts 6.","rationale":"The reader's verdict CONDITIONAL already flags the missing 'n ≥ k-1' restriction as a second issue, so this stress-test does not move the verdict. However, I regard the small-n counterexample as the single most load-bearing concern about the central claim: it directly falsifies Conjecture 2 as stated, and also invalidates Theorem 2 Part 3 for n < k-1, whereas the reader prioritized the 'by inspection' check in Lemma 5 for the 3-spider case of Theorem 3, which concerns the auxiliary δ* conjecture rather than the ex* central claim. The small-n issue is concrete, easily testable, and an unambiguous error in the statement. The paper's substantive contributions for n ≥ k-1 may still be correct, so a conditional acceptance with the required domain restriction is appropriate. The other concerns (Lemma 5's chromatic condition and the algebraic slip in Theorem 6 Part 4) are secondary and fixable, but they do not by themselves make the central claim false. Thus I agree with the reader that the paper needs revision before acceptance, but the most decisive reason is the explicit counterexample to the main conjecture.","tokens_in":13687,"tokens_out":14511,"duration_ms":130896,"concrete_test":"Compute ex*(Q_2,P_4) explicitly. Since Q_2 has exactly 4 edges, every subgraph has at most 4 edges, so ex*(Q_2,P_4) ≤ 4. Conjecture 2 predicts 6, giving an immediate contradiction. More generally, for any n and k with k-1 > n, the same total-edge bound refutes the conjecture; checking k = n+2 in Table 1 of the paper would confirm the typo. The fix is to add 'n ≥ k-1' to Conjecture 2, Theorem 2, and Theorem 3, and then verify the proofs only require this regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Conjecture 2, asserts ex*(Q_n,T) = (k-1)/2 · 2^n for every k-edge tree T and every n ∈ N. This is impossible whenever (k-1)/2 · 2^n exceeds the total number of edges of Q_n, namely n·2^{n-1}, i.e. whenever k-1 > n. For instance, take n=2 and T=P_4 (a path with 4 edges, so k=4). Then Q_2 has only 4 edges, so ex*(Q_2,P_4) ≤ 4, while the claimed value is (4-1)/2 · 2^2 = 6. Thus the conjecture is false as written. This also makes Theorem 2 Part 3 false for small n: a star with 4 edges has a vertex adjacent to 4 > 3/4·4 leaves, so the theorem would assert ex*(Q_2,K_{1,4}) = 6, again exceeding the total edges. The source of the problem is that the lower bound construction via Proposition 1 requires k-1 disjoint perfect matchings, i.e. n ≥ k-1, but the statements omit this hypothesis. The paper should either restrict all results and conjectures to n ≥ k-1, or state the asymptotically intended form. This is not a mere presentation issue: the literal central claim is false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines relative rainbow extremal numbers ex*(G,T) and the associated minimum-degree parameter δ*(G,T), and conjectures that for the hypercube Q_n these depend only on the number of edges of the tree T, with values (k−1)/2·2^n and k−1 respectively. The authors prove Theorem 2 (equality of ex*(Q_n,T) with this value for P3, P4, and trees with a vertex adjacent to more than 3/4 of its edges as leaves, plus a universal upper bound) and Theorem 3 (equality δ*(Q_n,T)=k−1 for paths, many-leaf trees, and certain spiders), relying on a general host-graph result (Theorem 6) and a greedy embedding lemma (Lemma 5). Lower bounds come from the 1-factorization of Q_n into coordinate matchings.","tokens_in":13961,"tokens_out":29317,"duration_ms":288790,"significance":"If the conjectures hold with the necessary dimension restriction n≥k−1, the paper identifies a natural host graph in which the rainbow extremal number regains the uniformity conjectured by Erdős–Sós. The proposed statements are clean and the partial verifications are nontrivial; the lower-bound construction by coordinate matchings is correct and elementary, and the universal bound ex*(Q_n,T)<(2k−1)2^n is a useful quantitative result. The paper also introduces a general framework for ex* on K3-free hosts that may be of independent interest. However, as written the central conjecture is false for small n, and one of the main proof lemmas has a serious gap.","major_comments":[{"comment":"The literal statements of Conjecture 2, Conjecture 3, and Theorem 2 parts 1–3 are false when n<k−1, because the claimed value (k−1)/2·2^n exceeds the total number of edges of Q_n, namely n·2^{n−1}. For example, for n=2 and T=P4 (k=4), Q2 has only 4 edges, so ex*(Q2,P4)≤4, whereas the conjecture predicts 6; likewise δ*(Q2,T)≤2<3=k−1. The lower-bound constructions require k−1 disjoint perfect matchings, so the dimension hypothesis n≥k−1 is necessary. Please add this restriction to all relevant conjectures and theorems, or explicitly state the intended asymptotic form.","section":"Conjecture 2 / Theorem 2"},{"comment":"The proof of Lemma 5's cycle-prevention step is not justified. It asserts that the coordinate assignment of the embedded edges is a proper vertex coloring of the auxiliary graph H, so that the chromatic hypothesis χ(H[P])≥ℓ+1 contradicts the fact that a cycle in Q_n uses at most ℓ coordinates. However, coordinates of two embedded edges are distinct only when the edges share a vertex in Q_n. For edges joined by H0 but disjoint in T, their images can share a coordinate when the corresponding path folds to a cycle (for instance, edges at even distance along the path can have identical coordinates in a folded cycle). Thus the coordinate map need not be a proper vertex coloring of H, and the claimed contradiction does not follow. Since Theorem 3 depends on Lemma 5, this gap must be repaired.","section":"Section 3.2, Lemma 5"},{"comment":"The greedy leaf-embedding count contains an algebraic error: (k−m)−2(k−m−ℓ) does not equal ℓ. The intended argument appears to be that after using the k−m unused neighbors of v0 and avoiding the colors of the k−m−ℓ non-adjacent edges, one has (k−m)−(k−m−ℓ)=ℓ valid choices; the extra factor 2 and the mention of 'coordinate' are spurious, since Theorem 6 is stated for general K_{2,r}-free graphs rather than for hypercubes. Please correct the calculation and clarify the counting.","section":"Section 4.2, proof of Theorem 6(4)"}],"minor_comments":[{"comment":"The statements of Theorem 2 and Theorem 3 should explicitly include the hypothesis n≥k−1; as written, the equalities are impossible when k−1>n because δ*(Q_n,T)≤n and ex*(Q_n,T)≤e(Q_n).","section":"Section 2 (Theorem 2 and Theorem 3)"},{"comment":"The verification of the four cases in Figure 5 is left to 'inspection'; please expand with a short written argument or a more detailed caption explaining the chromatic-number computations for each case.","section":"Section 3.2, proof of Theorem 3(4)"},{"comment":"The phrase 'color and coordinate' is confusing because coordinates are only defined for hypercubes, whereas Theorem 6 is stated for arbitrary K_{2,r}-free graphs; please remove the coordinate talk or explicitly restrict that step to Q_n.","section":"Section 4.2, proof of Theorem 6(4)"},{"comment":"The sentence 'While not a formal tool, these pictograms will help clarify the more delicate cases remaining in Theorem 3' appears twice; please delete the duplicate.","section":"Section 3.2"},{"comment":"Please verify reference [1]; the displayed year (1989) appears inconsistent with the volume of Israel Journal of Mathematics cited.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper contains promising ideas and several correct special cases, but the central conjecture is false as stated without the dimension restriction, and Lemma 5 — which underpins Theorem 3 — appears to have a genuine gap in the cycle-prevention argument. The authors should be asked to repair these issues before the paper can be accepted. The arithmetic slip in Theorem 6(4) is minor by comparison, but the other two points are load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a load-bearing flaw: Conjecture 2 as stated is false. For n=2 and T=P4 (k=4), the predicted value is 6 but Q2 has only 4 edges. The issue is that the lower-bound construction needs k-1 disjoint perfect matchings, i.e., n≥k-1, but the conjecture is written for all n. This is not a minor typo; it makes the central claim false. The fix is straightforward: restrict to n≥k-1 or state the asymptotic form.\n\nThat said, what is actually new is worth taking seriously. The paper introduces the relative rainbow extremal number on the hypercube, proposes that it satisfies an Erdős-Sós type statement, and proves the first exact results for small paths and trees with many leaves. The lower bounds from coordinate matchings are correct for n≥k-1. Lemma 5 is a clever greedy framework for upper bounds on δ*, and the verifications for paths, stars, and spiders are non-trivial. The universal upper bound ex*(Q_n,T)<(2k-1)2^n is a useful first step.\n\nThe soft spots are real but fixable. Besides the small-n issue, the many-leaves greedy count contains an algebraic error: (k-m)-2(k-m-ℓ) does not equal ℓ. And the proof for 3-spiders relies on 'inspection' of four cases, which is underdeveloped. There also seem to be typos in the statement of Theorem 2 part 1 (P3 value). None of these undermine the core idea, but they need attention.\n\nWho should read this? Extremal graph theorists and rainbow Turán people. The concept of relative rainbow extremal number is likely to be useful beyond this paper. It deserves a serious referee, but only after the authors fix the statements and the arithmetic. As written, I would not accept it; I would send it back for revision.","headline":"The central conjecture is false as stated for small n, but the fix is easy and the partial results are solid enough to justify a serious referee.","tokens_in":14503,"tokens_out":4176,"would_cite":true,"duration_ms":39339,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C05","05C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a rainbow Erdős–Sós conjecture for hypercubes: every $k$-edge tree should have rainbow extremal number $\\frac{k-1}{2}2^n$ in $Q_n$; it proves the exact value for $P_3$, $P_4$, and leaf-rich trees, and bounds all trees…","keywords":["rainbow extremal number","Erdős–Sós conjecture","hypercube","tree embedding","proper edge coloring","rainbow Turán problem","minimum degree"],"falsifier":"A direct refutation would be a properly edge-colored subgraph of $Q_n$ with $n\\ge k-1$, minimum degree $k$, and no rainbow copy of some $k$-edge tree $T$; that would disprove Conjecture 3 and imply Lemma 5's chromatic condition fails for $T$. Short of that, a computer search over all trees on $k\\le 8$ edges can check Lemma 5 directly: find a leaf-ordering and a $2\\ell$-edge path whose auxiliary induced subgraph has chromatic number at most $\\ell$, which would break the greedy-embedding proof for that tree. For Conjecture 2, the corresponding test is to find a properly colored subgraph of $Q_n$ with more than $\\frac{k-1}{2}2^n$ edges and no rainbow $T$.","tokens_in":13471,"feed_emoji":"🌈","tokens_out":16804,"duration_ms":146454,"temperature":0.7,"pith_summary":"The classical Erdős–Sós conjecture says that the ordinary extremal number of a $k$-edge tree $T$ depends only on $k$, not on the tree's shape. This paper asks whether the same structure-independence can hold for rainbow extremal numbers, where edges must be properly colored and no rainbow copy of $T$ may appear. The answer in complete graphs is no: a star and a path already have different rainbow extremal numbers, so the paper changes the host to the $n$-dimensional hypercube $Q_n$, which is 1-factorizable and therefore gives both a natural lower bound and a plausible setting for uniformity. The paper's central conjecture is that $\\mathrm{ex}^*(Q_n,T)=\\frac{k-1}{2}2^n$ for every $k$-edge tree $T$, and it verifies this exactly for the 2-edge and 3-edge paths and for any tree with a vertex adjacent to more than $\\frac{3}{4}k$ leaves, while proving the general upper bound $\\mathrm{ex}^*(Q_n,T)<(2k-1)2^n$. A second conjecture, that the rainbow minimum-degree parameter satisfies $\\delta^*(Q_n,T)=k-1$ for every tree, is verified for paths, long pendant paths, leaf-rich trees, and spiders with legs of even length or of length 3.","feed_headline":"For hypercubes, all k-edge trees may obey one rainbow bound","feed_subtitle":"The paper proves the star's exact bound for several tree families and bounds all trees within a factor of four.","key_machinery":"The central object is an auxiliary graph $H$ built from a leaf-ordering of the forbidden tree $T$. Given a leaf-ordering $x_0,\\dots,x_k$ of a $k$-edge tree with edges $e_i=x_{i'}x_i$, construct $H_0$ so that no edge $e_j$ has too many earlier $H_0$-neighbors, then form $H$ by adding edges between pairs of tree edges that share a vertex. Lemma 5 states that if every path of length $2\\ell$ in $T$ induces a subgraph of $H$ on its edge-vertices with chromatic number at least $\\ell+1$, then $\\delta^*(Q_n,T)\\le k-1$. The lemma works by greedily embedding $T$ into a properly colored minimum-degree-$k$ subgraph of $Q_n$, using the auxiliary graph to control which coordinates and colors are forbidden; a cycle in the embedding would let one color the path's edges with $\\ell$ coordinates, contradicting the chromatic condition. Most of Theorem 3 is a sequence of explicit constructions of $H_0$ for the relevant tree families, verifying this chromatic bottleneck.","core_discovery":"The paper's central proposal is a rainbow analogue of the Erdős–Sós conjecture with the hypercube as host: for every $k$-edge tree $T$, the relative rainbow extremal number $\\mathrm{ex}^*(Q_n,T)$ should equal $\\frac{k-1}{2}2^n$, the value already forced by a star. Because $Q_n$ decomposes into $n$ perfect matchings (one per coordinate), taking any $k-1$ of them gives a properly colored $(k-1)$-regular subgraph with no rainbow $k$-edge graph, supplying the lower bound; the content is matching upper bounds. The paper proves the equality for $T=P_3$, $T=P_4$, and for trees in which some vertex is adjacent to more than $\\frac{3}{4}k$ leaves, and a universal upper bound within a factor of about 4 for every tree. On the minimum-degree side, it proves $\\delta^*(Q_n,T)=k-1$ for paths, trees with a pendant path of at least $\\frac{3k-1}{4}$ edges, trees with at least $\\frac{k-1}{2}$ leaves, spiders with even-length legs, and spiders with legs of length 3. The heart of the upper-bound arguments is a greedy embedding lemma that reduces the problem to a purely combinatorial condition on the forbidden tree: each even-length path of edges must force its auxiliary graph to demand more colors than half the path length.","pith_inferences":["A natural testable extension is to run a computer search over all trees on small $k$, checking whether every leaf-ordering satisfies Lemma 5's chromatic condition; if a tree fails it, that indicates where the greedy method—and possibly the conjecture itself—needs a new idea.","Because Theorem 6 is proved for general $K_{2,r}$-free hosts rather than only $Q_n$, the exact value $\\frac{k-1}{2}|V(G)|$ for leaf-rich trees should extend to any $K_{2,r}$-free graph with $k-1$ disjoint perfect matchings, suggesting a broad family of hosts beyond the hypercube would satisfy a rainbow Erdős–Sós statement.","The proof for spiders with legs of length 3 (Theorem 3 Part 4) checks the finitely many cases of the auxiliary chromatic condition by inspection; an automated check of those four configurations would verify the last case directly.","The paper notes that adding a single diagonal edge to $Q_n$ destroys the lower-bound construction, which suggests the exact equality is fragile under small perturbations of the host; examining which subgraphs of the diagonal-augmented cube restore it could clarify what property of $Q_n$ is doing the work."],"forward_implications":["If Conjecture 2 holds, then in the hypercube the rainbow extremal number of every $k$-edge tree equals the ordinary extremal number of the star, $\\frac{k-1}{2}2^n$; the rainbow constraint would impose no extra structural cost for trees in this host.","The universal bound $\\mathrm{ex}^*(Q_n,T)<(2k-1)2^n$ gives the first linear-in-$|V(Q_n)|$ upper bound valid for every tree, with constant $2k-1$; this is the hypercube counterpart of the folklore bound $\\mathrm{ex}(n,T)\\le (k-1)n$ in complete graphs.","The minimum-degree equality $\\delta^*(Q_n,T)=k-1$ for the listed families means any rainbow-$T$-free properly colored subgraph of $Q_n$ must have a vertex of degree at most $k-1$, so the extremal subgraphs for minimum degree are exactly the unions of $k-1$ coordinate matchings.","The exact values for $P_3$ and $P_4$ establish the first path cases of the new conjecture: $\\mathrm{ex}^*(Q_n,P_3)=2^n$ and $\\mathrm{ex}^*(Q_n,P_4)=\\frac{3}{2}2^n$."],"supporting_citations":[{"why":"Introduces the rainbow extremal number $\\mathrm{ex}^*(n,F)$ and the inequality $\\mathrm{ex}(n,F)\\le \\mathrm{ex}^*(n,F)\\le \\mathrm{ex}(n,F)+o(n^2)$, the framework and measure the paper studies.","marker":"[16]"},{"why":"Shows $\\mathrm{ex}^*(n,P_k)\\ge \\frac{k}{2}n+O(1)$, the comparison demonstrating that in complete graphs the rainbow extremal number of a path differs from that of a star, motivating the hypercube host.","marker":"[14]"},{"why":"Constructs a proper edge-coloring of $K_{2p}$ with no rainbow Hamiltonian path; the paper uses this to show $\\delta^*(n,P_k)\\ge k$ and hence that the minimum-degree analogue also depends on tree structure.","marker":"[17]"}],"fun_headline_variants":["In hypercubes, all k-edge trees obey one rainbow bound","Rainbow Erdos-Sos on hypercubes: one bound for every tree","Hypercube rainbow extremal: all trees may share star's bound","Hypercube rainbow Erdos-Sos: a single bound for all trees"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every tree can be leaf-ordered so that, for every even-length path of edges, the auxiliary graph built from that path must need at least one more color than half the path's length; if any tree fails this condition, the greedy embedding proof of $\\delta^*(Q_n,T)=k-1$ collapses.","fun_headline_variants_meta":{"raw":{"variants":["In hypercubes, all k-edge trees obey one rainbow bound","Rainbow Erdos-Sos on hypercubes: one bound for every tree","Hypercube rainbow extremal: all trees may share star's bound","Hypercube rainbow Erdos-Sos: a single bound for all trees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001229,"raw_usage":{"total_tokens":5118,"prompt_tokens":1080,"completion_tokens":4038,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":696,"completion_tokens_details":{"reasoning_tokens":3959}},"tokens_in":696,"tokens_out":4038,"duration_ms":28768,"temperature":1.0,"reasoning_tokens":3959,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T20:05:52.968998+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct refutation would be a properly edge-colored subgraph of $Q_n$ with $n\\ge k-1$, minimum degree $k$, and no rainbow copy of some $k$-edge tree $T$; that would disprove Conjecture 3 and imply Lemma 5's chromatic condition fails for $T$. Short of that, a computer search over all trees on $k\\le 8$ edges can check Lemma 5 directly: find a leaf-ordering and a $2\\ell$-edge path whose auxiliary induced subgraph has chromatic number at most $\\ell$, which would break the greedy-embedding proof for that tree. For Conjecture 2, the corresponding test is to find a properly colored subgraph of $Q_n$ with more than $\\frac{k-1}{2}2^n$ edges and no rainbow $T$.","supporting_citations":[{"cited_title":"Rainbow tur´ an problems","cited_arxiv_id":null,"evidence_quote":"Introduces the rainbow extremal number $\\mathrm{ex}^*(n,F)$ and the inequality $\\mathrm{ex}(n,F)\\le \\mathrm{ex}^*(n,F)\\le \\mathrm{ex}(n,F)+o(n^2)$, the framework and measure the paper studies."},{"cited_title":"Lower bounds for rainbow tur´ an numbers of paths and other trees","cited_arxiv_id":null,"evidence_quote":"Shows $\\mathrm{ex}^*(n,P_k)\\ge \\frac{k}{2}n+O(1)$, the comparison demonstrating that in complete graphs the rainbow extremal number of a path differs from that of a star, motivating the hypercube host."},{"cited_title":"Maamoun and H","cited_arxiv_id":null,"evidence_quote":"Constructs a proper edge-coloring of $K_{2p}$ with no rainbow Hamiltonian path; the paper uses this to show $\\delta^*(n,P_k)\\ge k$ and hence that the minimum-degree analogue also depends on tree structure."}],"review_version":1}