Suns in triangle-free graphs of large chromatic number
Pith reviewed 2026-05-19 09:09 UTC · model grok-4.3
The pith
Triangle-free graphs of chromatic number at least 48 contain an induced t-sun for some t at least 5 or a 4-sun missing one pendant vertex.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For an integer t greater than or equal to 4, a t-sun is obtained from a t-vertex cycle by adding a degree-one neighbor to each cycle vertex. Every triangle-free graph of chromatic number at least 48 has an induced subgraph that is either a t-sun for some t at least 5 or a 4-sun with a single degree-one vertex deleted. In fact, for every ell at least 5 there exists a natural number c depending only on ell such that every triangle-free graph of chromatic number at least c has an induced subgraph that is either a t-sun for some t at least ell or a 4-sun with a single degree-one vertex deleted.
What carries the argument
The t-sun, a cycle of length t with one pendant vertex attached to each cycle vertex, which serves as the specific induced subgraph whose existence is forced once the chromatic number exceeds a finite threshold.
If this is right
- For every integer ell at least 5 there is a finite chromatic-number threshold guaranteeing an induced t-sun of size at least ell or the modified 4-sun.
- Triangle-free graphs with chromatic number at least 48 must contain an induced sun on five or more cycle vertices, or the near-4-sun variant.
- The original question of whether every sufficiently high-chromatic triangle-free graph contains some t-sun for t at least 4 is settled affirmatively except possibly for the exact 4-sun case.
- The same conclusion holds uniformly for all larger chromatic numbers once the threshold for a given ell is reached.
Where Pith is reading between the lines
- The result leaves open whether the chromatic-number threshold can be lowered below 48 while still forcing these suns.
- Similar forced-sun statements might be provable when the forbidden clique size is larger than 3 instead of exactly triangles.
- If the modified 4-sun is essential, it suggests that future structural theorems may need to treat the 4-sun as a separate base case.
- One could search for the smallest chromatic number that actually forces the stated suns by examining all triangle-free graphs up to that number.
Load-bearing premise
That once the chromatic number of a triangle-free graph passes a finite threshold, some Ramsey-type or extremal property must produce one of these particular induced sun subgraphs.
What would settle it
Exhibit a single triangle-free graph with chromatic number at least 48 that contains neither any induced t-sun for t at least 5 nor the 4-sun with one pendant vertex removed.
read the original abstract
For an integer $t\geq 4$, a $t$-sun is a graph obtained from a $t$-vertex cycle $C$ by adding a degree-one neighbor for each vertex of $C$. Trotignon asked whether every triangle-free graph of sufficiently large chromatic number has an induced subgraph that is a $t$-sun for some $t\geq 4$. This remains open, but we show that every triangle-free graph of chromatic number at least $48$ has an induced subgraph that is either a $t$-sun for some $t\geq 5$, or a $4$-sun with a single degree-one vertex deleted. In fact, we prove that for all $\ell\geq 5$, there exists $c=c(\ell)\in \mathbb{N}$ such that every triangle-free graph of chromatic number at least $c$ has an induced subgraph that is either a $t$-sun for some $t\geq \ell$, or a $4$-sun with a single degree-one vertex deleted.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that every triangle-free graph G with chromatic number at least 48 contains an induced subgraph that is either a t-sun for some t≥5 or a 4-sun with exactly one degree-one vertex deleted. It further shows that for every integer ℓ≥5 there exists a finite constant c=c(ℓ) such that any triangle-free graph with χ≥c contains an induced t-sun for some t≥ℓ or the indicated near-4-sun.
Significance. The result supplies a concrete finite threshold (48) and a general existence statement toward Trotignon's open question on induced suns in triangle-free graphs of large chromatic number. The argument proceeds by an explicit, finite case analysis that combines degeneracy bounds, neighborhood analysis in triangle-free graphs, and a Ramsey-type threshold; the constant 48 is obtained by iterating the reduction a fixed number of times. These features make the claim verifiable and give a clear structural consequence for high-chromatic triangle-free graphs.
minor comments (3)
- §1 (Introduction): the precise statement of the main theorem (the bound 48 and the general c(ℓ) result) should be displayed as a numbered theorem immediately after the definitions, rather than only in the abstract and the final paragraph of the introduction.
- §3 (Proof of the main reduction): the iteration count that produces the concrete bound 48 is described but not tabulated; adding a short table or explicit recurrence for the chromatic-number threshold after each reduction step would improve readability.
- Figure 1: the illustration of the 4-sun with a deleted pendant vertex is helpful, but the caption should explicitly label which vertex is removed and note that the resulting graph is still induced in the ambient triangle-free graph.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript, the recognition of its contribution toward Trotignon's question, and the recommendation of minor revision. The report accurately captures the main results, including the explicit threshold of 48 and the general existence statement for larger suns.
Circularity Check
No significant circularity
full rationale
The paper establishes an existence result for induced suns (or near-4-suns) in triangle-free graphs of chromatic number at least 48 (or some finite c(ℓ) for larger ℓ) via an explicit case analysis combining degeneracy bounds, neighborhood structure in triangle-free graphs, and iterated finite Ramsey-type thresholds. All steps reduce directly from the definitions of triangle-freeness and chromatic number without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations whose validity depends on the present work. The derivation is therefore self-contained and independent of its own outputs.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard definitions of chromatic number, induced subgraph, and triangle-free graph.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 1.2: triangle-free, 4-sunspot-free, t-sun-free (t≥5) graphs satisfy χ(G)≤43; extended in Theorem 1.3 via ℓ-safe flares and shortest-hole arguments.
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Construction of non-degenerate liberal flapless subgraphs via r-levelings and sector lemmas (Lemmas 2.3–2.6).
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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