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arxiv: 2604.11505 · v1 · submitted 2026-04-13 · 🧮 math.CO

Stability for the Anti-Ramsey Number of Matchings

Xuechun Zhang , Hongliang Lu This is my paper

Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification 🧮 math.CO
keywords anti-Ramsey numberrainbow matchingstability resultedge coloringmonochromatic subgraphcomplete graphmatching
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The pith

If an edge-colored K_n uses at least g(n,s) colors and has no rainbow matching of size s+2, then it contains either a monochromatic K_{n-s} or a monochromatic K_{n-2s-1} joined to the complement of K_{2s+1}.

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

The paper proves a stability theorem for the anti-Ramsey number of matchings. For n at least 2s plus 5, any edge-coloring of the complete graph on n vertices that uses at least g(n,s) colors yet avoids a rainbow matching of size s+2 must contain one of two large monochromatic subgraphs. A reader would care because such structural results classify the colorings that maximize colors without creating the forbidden rainbow object and often allow exact determination of the anti-Ramsey number itself. The work extends earlier bounds by replacing a pure numerical threshold with an explicit description of the extremal colorings.

Core claim

The central claim is that when n is at least 2s+5, an edge-coloring of K_n with at least g(n,s) colors, where g(n,s) equals the maximum of binom(n,2) minus binom(n-s+1,2) plus 5 and binom(2s-1,2) plus n plus 1, and with no rainbow matching of size s+2, must contain either a monochromatic complete graph on n-s vertices or a monochromatic join of a complete graph on n-2s-1 vertices with the complement of a complete graph on 2s+1 vertices.

What carries the argument

The stability threshold g(n,s) together with the forbidden rainbow matching of size s+2, which together force one of the two specified monochromatic subgraphs in the colored K_n.

If this is right

  • The extremal colorings that avoid rainbow matchings of size s+2 while using many colors are precisely those containing one of the two described monochromatic subgraphs.
  • The anti-Ramsey number ar(n, matching of size s+2) is at most g(n,s) under the given size condition on n.
  • Any coloring with more than g(n,s) colors must contain a rainbow matching of size s+2.
  • The two monochromatic structures identified are the only possible ways to reach the color threshold without creating the rainbow matching.

Where Pith is reading between the lines

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

  • The stability result can be used to confirm that g(n,s) is the exact anti-Ramsey number by checking the color counts inside the two monochromatic structures.
  • Similar stability arguments may apply directly to anti-Ramsey numbers for other forbidden rainbow graphs such as paths or cycles.
  • For fixed small s the structural conclusion can be checked exhaustively for moderate n to test the boundary cases of the theorem.

Load-bearing premise

The assumption that n is at least 2s+5 and that the coloring uses at least the specific threshold g(n,s) defined by the maximum of the two binomial expressions.

What would settle it

An edge-coloring of K_n with n at least 2s+5, at least g(n,s) colors, no rainbow matching of size s+2, and neither a monochromatic K_{n-s} nor a monochromatic K_{n-2s-1} vee overline{K_{2s+1}}.

read the original abstract

Let $n, r, s$ be three positive integers such that $n\geq 2s+5$. Let $K_r$ denote the complete graph of order $r$. Given a graph $F$, the anti-Ramsey number $ar(n,F)$ is defined as the minimum number $C$ such that any edge-coloring of $K_n$ with exactly $C$ colors contains a rainbow copy of $F$. Let $H$ be an edge-colored graph on $K_n$ with at least $g(n,s)$ colors, where \[ g(n,s)=\max\left\{ \binom{n}{2} - \binom{n - s + 1}{2} + 5, \binom{2s - 1}{2} + n + 1 \right\}. \] In this paper, we establish a stability type result for the anti-Ramsey number of matchings. Specifically, if $H$ does not have a rainbow matching of size $s+2$, then $H$ contains either a monochromatic complete graph $K_{n-s}$ or a monochromatic $K_{n - 2s - 1} \vee \overline{K_{2s + 1}}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves a stability result for the anti-Ramsey number of matchings: for integers n, s with n ≥ 2s + 5, any edge-coloring of K_n with at least g(n,s) colors (where g(n,s) is the maximum of binom(n,2) - binom(n-s+1,2) + 5 and binom(2s-1,2) + n + 1) that contains no rainbow matching of size s+2 must contain either a monochromatic K_{n-s} or a monochromatic K_{n-2s-1} ∨ overline{K_{2s+1}}.

Significance. If the result holds, it supplies a precise structural description of the extremal colorings avoiding large rainbow matchings, which strengthens the understanding of anti-Ramsey numbers ar(n, M_{s+2}) and may help determine exact values or related Turán-type problems in edge-colored graphs. The two-term form of g(n,s) directly encodes the two candidate extremal constructions.

major comments (2)
  1. [Main theorem / §2] The hypothesis n ≥ 2s + 5 and the exact max-of-two-binomials definition of g(n,s) are load-bearing; §2 (or the main theorem statement) should contain an explicit verification or exhaustive case check for the boundary instance s=2, n=9 to confirm that no hybrid coloring using g(9,2) colors avoids both a rainbow 4-matching and the two forbidden monochromatic subgraphs.
  2. [Proof of Theorem 1.1] In the proof of the stability statement (likely §3 or §4), the case division between the two summands of g(n,s) must be shown to cover all colorings when n is only a few units larger than 2s+5; the additive +5 in the first term may not absorb all possible overlaps between the two extremal constructions, and a concrete counter-example search or additional lemma is needed if the separation fails.
minor comments (2)
  1. [Introduction / Notation] Define the join notation ∨ and the complement overline{K_m} explicitly at first use, and ensure the statement of the main result matches the abstract verbatim.
  2. [Introduction] Add a short table or paragraph in the introduction comparing g(n,s) to previously known lower and upper bounds on ar(n, M_{s+2}).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions regarding the boundary cases and case divisions in our stability result. We address each major comment below and will revise the manuscript to enhance clarity without altering the core proof.

read point-by-point responses

  1. Referee: [Main theorem / §2] The hypothesis n ≥ 2s + 5 and the exact max-of-two-binomials definition of g(n,s) are load-bearing; §2 (or the main theorem statement) should contain an explicit verification or exhaustive case check for the boundary instance s=2, n=9 to confirm that no hybrid coloring using g(9,2) colors avoids both a rainbow 4-matching and the two forbidden monochromatic subgraphs.

    Authors: We agree that an explicit verification for the boundary case s=2, n=9 would improve readability, even though the general argument applies directly since 9 = 2·2 + 5. In the revised version we will insert a short remark or subsection in §2 that specializes the proof techniques to this instance, confirming by direct case analysis on color distributions that no hybrid coloring with exactly g(9,2) colors can avoid both a rainbow 4-matching and the two monochromatic subgraphs. revision: yes

  2. Referee: [Proof of Theorem 1.1] In the proof of the stability statement (likely §3 or §4), the case division between the two summands of g(n,s) must be shown to cover all colorings when n is only a few units larger than 2s+5; the additive +5 in the first term may not absorb all possible overlaps between the two extremal constructions, and a concrete counter-example search or additional lemma is needed if the separation fails.

    Authors: The proof already separates colorings according to which term of g(n,s) is active and uses the +5 precisely to guarantee that, for all n ≥ 2s+5, any coloring meeting the threshold cannot be a hybrid that evades both extremal structures. The arguments in §§3–4 are uniform and do not rely on n being much larger than 2s+5. Nevertheless, to address the concern explicitly, we will add a brief clarifying paragraph (or short lemma) immediately after the definition of g(n,s) that verifies the separation holds for the first few values n = 2s+5, 2s+6, … without overlap; no separate computational search is required because the existing analytic estimates already cover these cases. revision: partial

Circularity Check

0 steps flagged

No circularity: explicit threshold definition and structural conclusion are independent of fitted parameters or self-referential chains.

full rationale

The paper defines g(n,s) explicitly via two binomial expressions and proves the stability implication directly from the assumption of at least g(n,s) colors without a rainbow (s+2)-matching. No derivation step reduces the claimed monochromatic subgraphs to a redefinition of g, a fitted input, or a load-bearing self-citation; the argument proceeds by combinatorial case analysis on colorings avoiding the rainbow matching. The n ≥ 2s+5 hypothesis is an explicit domain restriction rather than a derived quantity, and the result remains self-contained against the standard definition of anti-Ramsey numbers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions of rainbow and monochromatic subgraphs plus the combinatorial threshold g(n,s); no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • standard math Standard axioms of graph theory including definitions of edge-colorings, rainbow copies, and monochromatic subgraphs.
    Invoked throughout the statement of the anti-Ramsey number and the structural conclusion.

pith-pipeline@v0.9.0 · 5516 in / 1240 out tokens · 38314 ms · 2026-05-10T15:55:05.063750+00:00 · methodology

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Reference graph

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