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The paper proves a sharp bound on properly colored K4s in 3-edge-colored graphs and shows the extremal graph is always a balanced blowup of a properly colored K4.

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A graph with R red, G green, B blue edges contains at most ¼(RGB)^{2/3} properly colored K4s, with equality only for balanced blowups of a properly colored K4; the known rainbow-triangle bound √(2RGB) receives new flag-algebra, counting, and entropy proofs.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Theorem 1.3 is a solid, sharply proved new bound; but the advertised k≥4 generalization and Theorem 1.4 are not currently supported, and the abstract overstates what the paper proves. the 2 major comments →

arxiv 2511.21061 v3 pith:BWDVCKKU submitted 2025-11-26 math.CO

Density of rainbow triangles and properly colored $K_4$'s

J\'ozsef Balogh , Peter Bradshaw , Ramon I. Garcia , Bernard Lidick\'y This is my paper
classification math.CO MSC 05C3505C15
keywords properly colored K4rainbow trianglesedge-colored graphsextremal graph theoryflag algebrasentropy methodblowup constructionCauchy-Schwarz inequality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an exact extremal statement: in any graph with R red, G green, and B blue edges, the number K of properly 3-edge-colored K4s is at most one quarter of (RGB)^(2/3), and the stronger inequality K ≤ ¼ min{RG, GB, RB} also holds. If equality is achieved in the geometric-mean form, the graph must be a balanced blowup of a properly colored K4, possibly with isolated vertices. This pins the maximum count and the extremal structure entirely in terms of the three color-class sizes, a rare outcome for subgraph-counting problems. The same machinery gives computer-free flag-algebra, elementary counting, and shorter entropy proofs of the known rainbow-triangle bound T ≤ √(2RGB), plus a uniqueness theorem for its equality case, and a sharp bound for fixed rainbow 6-colorings of K4. A sympathetic reader should come away with a complete, structural answer: edge counts alone determine the maximum and the extremal graphs.

Core claim

Theorem 1.3 is the central new result: a 3-edge-colored graph with R red, G green, and B blue edges has at most ¼(RGB)^(2/3) properly colored K4s, and if the bound is met with positive count, the non-isolated part of the graph is a balanced blowup of a properly colored K4. The proof derives the stronger bound 4K ≤ min{RG, GB, RB} by counting ordered 4-tuples that form a double-rainbow configuration and injecting them into pairs of green and blue edges (Lemma 2.3), then applying Cauchy-Schwarz; color permutations yield the minimum. Equality in the geometric-mean form forces all inequalities to be tight, which translates into the structural conditions of Lemma 2.4—every pair of edges of distin

What carries the argument

The argument rests on an injection lemma: ordered 4-tuples (u,v,x,y) with uv red, ux and uy blue, vx and vy green map injectively to unordered pairs consisting of one green edge and one blue edge, so their number is at most G·B. Summing over red edges and applying Cauchy-Schwarz gives 4K ≤ G·B, and permuting colors yields 4K ≤ min{RG, GB, RB}, whose geometric mean is the (RGB)^(2/3) bound. Uniqueness is carried by a structural lemma that characterizes any graph satisfying (a) every pair of distinct-colored edges is joined by a third-color edge and (b) all vertices have equal red/green/blue degrees as a balanced blowup of a properly colored K4. The flag-algebra proofs are translated to finite

Load-bearing premise

The paper's rainbow-K4 bound depends on an unverified six-color analogue of a flag-algebra inequality that is asserted by analogy and never checked explicitly, whereas the central properly-colored-K4 theorem rests on a fully verified injection argument.

What would settle it

A single 3-edge-colored graph with R, G, B edges and more than ¼(RGB)^(2/3) properly colored K4s would refute Theorem 1.3, as would an equality case whose non-isolated part is not a balanced blowup of a properly colored K4; an exhaustive computer search over small graphs with fixed R, G, B could look for both.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Exact counting: any red/green/blue graph has at most ¼(RGB)^(2/3) properly colored K4s, and the balanced blowup of a properly colored K4 attains the bound, so it is best possible.
  • Rigidity of equality: if K = ¼(RGB)^(2/3) > 0, the graph is forced to be a balanced blowup of a properly colored K4 plus isolated vertices; there are no other extremal examples.
  • New proofs for the rainbow triangle bound T ≤ √(2RGB): the paper supplies a computer-free flag-algebra proof, an elementary injection-based counting proof, and a shorter entropy proof, plus a uniqueness statement for the equality case.
  • For any fixed rainbow 6-edge-coloring of K4, the number of its copies H satisfies H ≤ (∏ C_i)^(1/3), and this is sharp for blowups of that coloring; if the abstract's generalization holds, the same form extends to fixed rainbow K_k for k ≥ 4.
  • The stronger inequality K ≤ ¼ min{RG, GB, RB} means the count is controlled by the two smallest color classes, not merely by the product of all three.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The injection philosophy likely generalizes: any fixed pattern in an edge-colored graph that admits an injective map from its ordered-tuple realizations into pairs of edges of two colors will satisfy an analogous product bound, giving a template for other small colored subgraphs.
  • Because properly colored K4s correspond to tetrahedra in the multijoint/generically induced configuration setting, the theorem supplies a sharp constant for a colored incidence-geometry problem; the uncolored tetrahedron problem (Problem 1 in the paper) remains open and may need different methods.
  • A testable extension suggested by the paper's Question 2 is whether the rainbow (not fixed) K4 count also obeys H ≤ (∏ C_i)^(1/3); the paper proves this only for a fixed rainbow coloring, and finding an injective counting argument for the rainbow case would settle the question.
  • The abstract's promised k ≥ 4 rainbow-K_k bounds rely on an unverified '6-color analogue' of a flag-algebra inequality; verifying that inequality on all 6-vertex 6-edge-colored graphs (or finding a counterexample) would either support or refute that broader claim, while the main K4 theorem is not affected.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies extremal bounds on rainbow triangles and properly 3-edge-colored K4's in a 3-edge-colored graph, expressed in terms of the numbers R, G, B of red, green, and blue edges. It gives a computer-free flag-algebra proof of the Chao–Yu bound T^2 ≤ 2RGB, translates that proof into an elementary counting proof and an entropy proof, and proves uniqueness of the extremal blowup construction. The main new result is Theorem 1.3: the number K of properly colored K4's satisfies K ≤ ¼ min{RG, GB, RB} ≤ ¼(RGB)^{2/3}, with equality forcing a balanced blowup of a properly colored K4 plus isolated vertices. The paper also states Theorem 1.4, a six-color analogue for a fixed rainbow coloring of K4, and claims a general result for every k ≥ 4 in the abstract.

Significance. If Theorem 1.3 is correct, it is a significant and attractive result: the extremal number of properly colored K4's is exactly determined by the sizes of the three color classes, and the extremal structure is unique. The elementary proof via Lemma 2.3 and the Cauchy–Schwarz chains is transparent and gives a genuinely computer-free argument, which is a methodological strength. The entropy and counting proofs also add value. However, Theorem 1.4 and the k ≥ 4 claims advertised in the abstract are not actually proved in the manuscript; as it stands, the paper's scope exceeds its content, and this needs to be resolved before publication.

major comments (2)
  1. [Section 4, proof of Theorem 1.4, chain leading to (12)] The proof invokes 'a 6-color analogue of (3)' without stating or proving it. This inequality is load-bearing: without it, the displayed chain from the Cauchy–Schwarz step to (12) cannot be verified, and no elementary cross-check is provided. The authors must either state and prove this analogue explicitly or give a reference for it. In addition, the final conclusion of Section 4 says 'H≤∏ C_i'; the correct conclusion from the preceding line is H≤(∏ C_i)^{1/3}. The cube root is dropped, and as written the claimed statement is not the sharp bound stated in Theorem 1.4.
  2. [Abstract and Section 4] The abstract promises that for every k≥4 and a fixed rainbow coloring of K_k, a sharp upper bound is given, relying on a 'new flag-algebra version of Hölder's inequality.' No such k≥4 theorem appears in the body, and no version of Hölder's inequality is stated. The only rainbow-K4 result, Theorem 1.4, is itself unsupported because of the missing 6-color analogue. The authors should either include the missing statements and proofs or explicitly restrict the claims in the abstract and introduction.
minor comments (5)
  1. [Section 3, proof of Theorem 1.3] The sentence 'we assume that each edge of Γ belongs to at least one properly colored K4' is not justified at that point. The same deletion argument used in the proof of Theorem 1.2 works, but it should be stated explicitly rather than left implicit.
  2. [Section 4, last line] As noted in the major comments, the cube root is missing in 'H≤ Q6 i=1 Ci'. This is presumably a typographical error, but it is important to fix.
  3. [Question 2, Introduction] The displayed bound 'H≤ 3 pQ i Ci' is ambiguous; it should be written as H≤(∏ C_i)^{1/3} to be clear.
  4. [Section 4, derivation of (12)] The step 'By symmetry we get = ...^{1/3} × ...' is cryptic. Please spell out how the geometric mean of the three pairwise bounds is obtained, since this is not immediate from the displayed chain.
  5. [Section 2 opening] The paper refers to an appendix in the arXiv version for flag-algebra background, but that appendix is not present in the submitted manuscript. For a self-contained journal submission, the authors should either include the appendix or cite a published reference for the needed background.

Circularity Check

0 steps flagged

No circularity found: Theorem 1.3 is proved by an independent elementary argument; Section 4 contains an unsupported analogue, but this is a missing proof, not a circular reduction.

full rationale

The central derivation chain is self-contained. Theorem 1.3 is proved via Claim 3.2, whose proof is elementary: the paper defines quantities dK(uv), d-(uv), d+(uv), applies Cauchy–Schwarz, and invokes Lemma 2.3, which is proved independently in the paper by an injective map from S to S′. No parameter is fitted, and the bound K ≤ 1/4 min{RG,GB,RB} is not assumed in any form. Theorem 1.1 is likewise re-derived from scratch by Lemma 2.3 plus Cauchy–Schwarz, and separately by an entropy argument using the same lemma; the Chao–Yu theorem is cited only as prior work, not used as an input. The flag-algebra framework is Razborov's published theory [11], and the internal inequalities are either derived in the text or, in the case of Section 4, merely asserted. The only genuinely unsupported load-bearing step is the '6-color analogue of (3)' in the proof of Theorem 1.4, which is neither stated nor proved, and the final displayed simplification drops the cube root ('which simplifies to H ≤ ∏ Ci' after deriving a cube-root bound). These are correctness gaps, not circularity: nothing in the proof reduces to the theorem being proved by construction, and no self-citation carries the argument. The stability sketch in Section 5 is also explicitly omitted, but again that is an omitted proof, not a circular dependency. Thus the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central derivations are axiom-lean. No free parameters: the 'd' appearing in Lemma 2.4 is derived from the equality conditions of Cauchy–Schwarz, not chosen. No invented entities. The imported axioms are Razborov's flag-algebra theory (used in Lemma 2.1, Lemma 3.1, and §4) and standard entropy facts in the entropy proofs. One ad-hoc unproved inequality is introduced: the '6-color analogue of (3)' in Theorem 1.4's proof, on which that theorem rests alone. The 3-color inequalities (3) and (7) have verbal verifications and are corroborated by the elementary proofs of the same results, so they are not counted as axioms.

axioms (3)
  • standard math Razborov's flag-algebra framework: convergent sequences of finite 3-edge-colored graphs have limits in Hom⁺(A,R); flag Cauchy–Schwarz inequality JF×GK ≤ √(JF²K·JG²K) [11, Thm 3.14].
    Invoked in Lemma 2.1, Lemma 3.1, and the proof of Theorem 1.4; standard published theory, not machine-checked in this paper.
  • standard math Standard entropy facts: chain rule, H(X,Y|Z) ≤ H(X|Z)+H(Y|Z), H(X|Y,Z) ≤ H(X|Z), and entropy of a uniform distribution over N elements is log₂N.
    Used in the entropy proofs of Theorem 1.1 and Claim 3.2 (Sections 2 and 3).
  • ad hoc to paper The '6-color analogue of (3)' flag-algebra inequality in the proof of Theorem 1.4 (Section 4, leading to (12)): asserted without explicit enumeration or proof, with no elementary cross-check.
    This inequality is specific to this paper's proof of Theorem 1.4 and is load-bearing for that theorem; its verification is delegated to analogy with (3). It is also the type of tool the abstract's k≥4/Hölder claims would require.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Density of rainbow triangles and properly colored $K_4$'s." pith.science (2026). https://pith.science/paper/BWDVCKKU

@misc{pith2026251121061,
  author       = {Pith},
  title        = {Pith review of: Density of rainbow triangles and properly colored $K_4$'s},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWDVCKKU}},
  note         = {Machine review of arXiv:2511.21061}
}
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abstract

We establish a sharp upper bound on the number of properly $3$-edge-colored $K_4$'s in graphs with $R$ red, $G$ green and $B$ blue edges. We give a computer-free flag-algebra proof of this bound, and we also convert our proof into a classical counting proof and an entropy proof. Additionally, for every $k\ge 4$, for a fixed rainbow coloring $F$ of a complete graph $K_k$, we give a sharp upper bound on the number copies of $F$ in a $\binom{k}{2}$-edge-colored graph. Our proof of this result relies on a new flag-algebra version of H\"older's inequality. We also give a computer-free flag-algebra proof of the fact that a graph with $R$ red, $G$ green, and $B$ blue edges has at most $\sqrt{2 RGB}$ rainbow triangles, which was originally proven by T.-W. Chao and H.-H. H. Yu using the entropy method. We also provide an even shorter entropy proof of their result.

Figures

Figures reproduced from arXiv: 2511.21061 by Bernard Lidick\'y, J\'ozsef Balogh, Peter Bradshaw, Ramon I. Garcia.

Figure 1
Figure 1. Figure 1: (a) A blowup of a properly 3-edge-colored [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two iterated constructions of a 6-edge-colored complete graph where rainbow [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Members of sets S and S ′ from Lemma 2.3. Proof. To prove the lemma, we show that the function f : S → S ′ mapping (u, v, x, y) 7→ ({u, x}, {v, y}) is injective. Indeed, choose an arbitrary element (g, b) in the image of f. We show that the vertices u, v, x, y can be uniquely determined from (g, b). Case (i): Suppose that g and b share an endpoint z. Then, the set {u, v, x, y} contains at most three distin… view at source ↗
Figure 4
Figure 4. Figure 4: A drawing of Γ[v1, v2, v3, v4] and the graph obtained by adding the resampled vertices v ′ 1 , v′ 2 . 4 Proof of Theorem 1.4 Proof of Theorem 1.4. An application of the Cauchy-Schwarz inequality and a 6-color analogue of (3) give = 24 · s 1 2 1 2 { ≤ 24 · s 1 2 × 1 2 { ≤ 24 · vuut t 1 2 2 | · t 1 2 2 | ≤ 24 · vuut 1 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

discussion (0)

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.