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A linear upper bound on zero-sum Ramsey numbers of d-degenerate graphs in mathbb{Z}_p
Pith reviewed 2026-05-10 16:35 UTC · model grok-4.3
The pith
For d-degenerate graphs with enough edges the zero-sum Ramsey number over Z_p is at most n plus (3+3d)p.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If G is a d-degenerate graph on n vertices and m edges, then R(G, Z_p) ≤ n + (3+3d)p so long as m ≥ 2 p d (d+1)^2, p divides m, and 2d < p. This generalizes a result by Colucci and D'Emidio on 1-degenerate graphs.
What carries the argument
The zero-sum Ramsey number R(G, Z_p), the smallest ell such that every Z_p-edge-coloring of K_ell contains a copy of G with edge-color sum zero; the proof uses d-degeneracy to guarantee such a copy exists inside the stated linear window.
If this is right
- The linear bound applies to every d-degenerate graph that meets the edge threshold, including many bipartite graphs and outerplanar graphs.
- The additive term depends only on d and p, not on n or m beyond the threshold.
- The result recovers the earlier bound for 1-degenerate graphs as the special case d=1.
- The bound requires p to be a prime larger than twice the degeneracy.
Where Pith is reading between the lines
- The edge-count threshold may be an artifact of the current proof and could perhaps be removed or lowered.
- Analogous linear bounds might hold when the colors come from other abelian groups instead of Z_p.
- Computational checks for small fixed d and p could reveal whether the coefficient 3+3d is tight.
Load-bearing premise
The graph must have at least 2 p d (d+1)^2 edges and that number must be divisible by p.
What would settle it
A concrete d-degenerate graph with m edges where m is below 2 p d (d+1)^2 or p does not divide m, yet whose zero-sum Ramsey number exceeds n + (3+3d)p for some prime p > 2d.
read the original abstract
Let $p$ be a prime number and let $G$ be a graph on $n$ vertices and $m$ edges. The zero-sum Ramsey number of $G$ over $\mathbb{Z}_p$, denoted by $R(G, \mathbb{Z}_p)$, is the minimum $\ell\in \mathbb{N}$ such that for any edge-coloring $c:E(K_\ell)\to\mathbb{Z}_p$, there is a subgraph $G'\subset K_\ell$ isomorphic to $G$ and satisfying $\sum_{e\in E(G')}c(e)=0$. We prove that if $G$ is a $d$-degenerate graph, then $R(G, \mathbb{Z}_p)\leq n + (3+3d)p$ so long as $m\geq 2pd(d+1)^2$, $p$ divides $m$, and $2d<p$. This generalizes a result by Colucci and D'Emidio on $1$-degenerate graphs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that if G is a d-degenerate graph on n vertices with m edges satisfying m ≥ 2pd(d+1)^2, p divides m, and 2d < p (p prime), then the zero-sum Ramsey number R(G, Z_p) satisfies R(G, Z_p) ≤ n + (3 + 3d)p. This extends the d=1 case previously obtained by Colucci and D'Emidio.
Significance. If the proof is correct, the result supplies an explicit linear upper bound (in both n and p) for zero-sum Ramsey numbers of d-degenerate graphs under natural density and divisibility hypotheses. Such bounds are of interest in Ramsey theory and additive combinatorics; the explicit constants and the generalization from the matching case constitute a modest but concrete advance.
minor comments (3)
- [Introduction / Theorem 1] The abstract states the three conditions explicitly, but the introduction or statement of the main theorem should clarify whether these conditions are necessary for the linear bound or merely sufficient for the current proof technique.
- [Section 1] Notation for the zero-sum Ramsey number R(G, Z_p) is defined clearly in the abstract; ensure the same definition appears verbatim in the first section of the manuscript and that the isomorphism type of G' is stated unambiguously.
- [Introduction] The constant 3 + 3d appears without derivation in the abstract; a brief remark in the introduction on how the factor 3 arises (e.g., from an application of the Cauchy-Davenport theorem or a greedy embedding) would help readers.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our work, the accurate summary of the main theorem, and the recommendation of minor revision. We appreciate the recognition that the result provides an explicit linear upper bound and generalizes the d=1 case.
Circularity Check
No significant circularity; bound is conditional and self-contained
full rationale
The paper states an explicit conditional theorem: for d-degenerate G with m edges satisfying m ≥ 2pd(d+1)^2, p|m and 2d < p, the zero-sum Ramsey number R(G, Z_p) ≤ n + (3+3d)p. This is presented as a direct upper-bound proof that generalizes the d=1 case of Colucci and D'Emidio without invoking fitted parameters, self-definitional reductions, or load-bearing self-citations. The edge-count and divisibility restrictions are transparently included in the hypothesis, so the claimed linear bound does not reduce to a tautology or to its own inputs by construction. No equations or ansatzes in the abstract or stated result exhibit the enumerated circularity patterns.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Z_p is a field (prime order) and edge sums are taken in this group
- domain assumption A graph is d-degenerate if every subgraph has minimum degree at most d
Reference graph
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