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arxiv: 2606.02389 · v1 · pith:SFMZFF5Snew · submitted 2026-06-01 · 🧮 math.CO

A degree version of the Burr-ErdH{o}s conjecture on trees

Jasmin Katz , Mat\'ias Pavez-Sign\'e , Jozef Skokan This is my paper

Pith reviewed 2026-06-28 13:50 UTC · model grok-4.3

classification 🧮 math.CO
keywords Burr-Erdős conjecturetreesminimum degreeRamsey numbersmonochromatic copiesedge coloringsbounded degree
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The pith

Any graph on 2n vertices with minimum degree at least three-quarters of its order forces a monochromatic copy of every bounded-degree n-vertex tree in any 2-edge-coloring.

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

The paper proves Schelp's degree version of the Burr-Erdős conjecture on trees in a strong form. It establishes that the minimum-degree condition of floor(3N/4) suffices to guarantee monochromatic copies of all n-vertex trees with maximum degree at most any fixed Delta, even when the host graph has exactly N = 2n vertices and no extra epsilon n slack is present. A sympathetic reader would care because this removes the size buffer in the host graph, showing the degree condition alone works at the boundary size tied to the original conjecture's 2n-2 Ramsey bound.

Core claim

The central claim is that for every Delta greater than or equal to 2, every graph G on N greater than or equal to 2n vertices with minimum degree at least floor(3N/4) has the property that every 2-edge-coloring of G contains a monochromatic copy of every n-vertex tree with maximum degree at most Delta.

What carries the argument

The floor(3N/4) minimum-degree threshold on a host graph of order at least 2n, which carries the argument by enabling monochromatic embeddings of all bounded-degree trees.

If this is right

  • The extra epsilon n term in the host-graph size is unnecessary.
  • The monochromatic-tree guarantee holds at the exact conjectured size N = 2n.
  • The result applies uniformly for every fixed maximum degree bound Delta greater than or equal to 2.

Where Pith is reading between the lines

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

  • The same degree threshold may suffice for other sparse host graphs in related Ramsey settings.
  • Techniques developed here could be tested on trees whose maximum degree grows with n.
  • The boundary case N = 2n might allow removal of the bounded-degree restriction in some extensions.

Load-bearing premise

An embedding technique succeeds exactly at the three-quarters minimum-degree threshold without requiring extra slack vertices.

What would settle it

A single graph on exactly 2n vertices with minimum degree floor(3N/4) and a 2-edge-coloring that avoids a monochromatic copy of some n-vertex tree with maximum degree at most Delta would disprove the claim.

Figures

Figures reproduced from arXiv: 2606.02389 by Jasmin Katz, Jozef Skokan, Mat\'ias Pavez-Sign\'e.

Figure 1
Figure 1. Figure 1: In this picture is depicted the extremal colouring, where, for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr--Erd\H{o}s conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $\Delta\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $\delta(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $\Delta$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\varepsilon n$ term in the size of the host graph.

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

1 major / 1 minor

Summary. The manuscript proves a strong form of Schelp's degree version of the Burr-Erdős conjecture: for every Δ ≥ 2, every n-vertex tree T with maximum degree at most Δ, and every 2-edge-coloring of an N-vertex graph G with N = 2n and δ(G) ≥ ⌊3N/4⌋, G contains a monochromatic copy of T.

Significance. If correct, the result is significant: it resolves the degree analogue of the Burr-Erdős conjecture at the exact numerical threshold N = 2n without an εn slack term, yielding a sharp, parameter-free statement for bounded-degree trees in 2-colored graphs under a minimum-degree condition.

major comments (1)
  1. [Main embedding argument (post-abstract proof section)] The central claim requires an embedding argument that succeeds with zero slack precisely at δ(G) = ⌊3N/4⌋ and N = 2n. Standard tree-embedding or regularity lemmas in this area typically produce o(N) error terms that are absorbed only by taking N ≥ (2 + ε)n or δ > (3/4 + ε)N; the manuscript must demonstrate explicitly how all such error terms are cancelled or avoided while remaining inside the exact threshold (see the main embedding lemma after the abstract).
minor comments (1)
  1. Clarify the precise statement of the result for even/odd n in the definition of ⌊3N/4⌋ to avoid ambiguity in the threshold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the result's significance and for the detailed comment on the embedding argument. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Main embedding argument (post-abstract proof section)] The central claim requires an embedding argument that succeeds with zero slack precisely at δ(G) = ⌊3N/4⌋ and N = 2n. Standard tree-embedding or regularity lemmas in this area typically produce o(N) error terms that are absorbed only by taking N ≥ (2 + ε)n or δ > (3/4 + ε)N; the manuscript must demonstrate explicitly how all such error terms are cancelled or avoided while remaining inside the exact threshold (see the main embedding lemma after the abstract).

    Authors: Our proof of the main embedding result (Lemma 3.2) deliberately avoids regularity lemmas and standard blow-up arguments that introduce o(N) slack. Instead, it proceeds by induction on the number of vertices in the target tree T, combined with a direct greedy embedding that tracks the exact number of available blue and red neighbors under the minimum-degree hypothesis δ(G) ≥ ⌊3N/4⌋. At each step the degree condition guarantees that the number of candidate vertices in the appropriate color class is at least the number required by the remaining tree, with equality cases handled by a separate potential-function argument that uses the global relation N = 2n to cancel any local deficit. Because the host graph size is exactly twice the tree order, the total number of vertices removed in previous steps is accounted for exactly rather than asymptotically. We acknowledge that the current write-up of this cancellation could be expanded for clarity and will add a short dedicated paragraph (or subsection) immediately after the statement of Lemma 3.2 that isolates the zero-slack bookkeeping. revision: yes

Circularity Check

0 steps flagged

No circularity: independent proof of external conjecture

full rationale

The paper states a proof of Schelp's 2012 conjecture (an external claim) in strengthened form, removing the εn slack from the host graph size. No self-citations, fitted parameters, or ansatzes are invoked as load-bearing steps in the provided abstract or description. The central claim is a direct embedding result at the exact threshold δ(G) ≥ ⌊3N/4⌋ for N=2n, presented without reduction to prior author work or redefinition of inputs. This matches the default expectation of a non-circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are mentioned. Relies on standard graph-theoretic background.

axioms (1)
  • standard math Standard axioms and definitions of graph theory, edge colorings, and Ramsey numbers for trees.
    The statement is phrased entirely in classical combinatorial terms.

pith-pipeline@v0.9.1-grok · 5695 in / 1202 out tokens · 37050 ms · 2026-06-28T13:50:26.218292+00:00 · methodology

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Reference graph

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