New Upper Bound for the Edge Folkman Number Fe(3,5;13)

Nikolay Kolev

arxiv: 0806.1403 · v1 · pith:SGRS5FWZnew · submitted 2008-06-09 · 🧮 math.CO

New Upper Bound for the Edge Folkman Number Fe(3,5;13)

Nikolay Kolev This is my paper

Pith reviewed 2026-05-19 04:57 UTC · model grok-4.3

classification 🧮 math.CO

keywords edge Folkman numberRamsey theorygraph arrowingclique avoidanceupper boundvertex order 21

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The pith

The edge Folkman number Fe(3,5;13) is at most 21.

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

The paper establishes a new upper bound by exhibiting a graph on 21 vertices that satisfies the Folkman arrowing property for parameters (3,5) while avoiding a clique of size 13. This shows that any graph on 22 or more vertices that arrows (3,5) must contain a K_13 only if smaller examples fail, tightening the known range for this Ramsey-type number. A reader would care because Folkman numbers measure the minimal size at which local coloring conditions force global cliques, and concrete bounds help map the boundary between forced and avoidable structures in graph theory.

Core claim

The author proves that Fe(3,5;13) ≤ 21 by constructing or verifying the existence of a 21-vertex graph G such that any 2-edge-coloring of G without a monochromatic triangle in color 1 or a monochromatic K_5 in color 2 must contain a monochromatic K_13 in one of the colors, while G itself contains no K_13.

What carries the argument

A 21-vertex graph that arrows (3,5) without containing K_13, serving as a witness for the upper bound on the minimal order of any such arrowing graph.

If this is right

The minimal order of any graph forcing a monochromatic K_13 under (3,5)-arrowing is now known to lie between previous lower bounds and 21.
Any future construction attempting to show Fe(3,5;13) < 21 must produce a strictly smaller witness graph.
Computational searches for Folkman graphs can now target the interval up to 21 vertices as the relevant search space.

Where Pith is reading between the lines

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

Similar explicit constructions on small vertex sets may tighten bounds for other edge Folkman numbers Fe(k,l;r) with r near the Ramsey threshold.
If the 21-vertex witness is vertex-transitive or has high symmetry, it could suggest algebraic or geometric constructions for related arrowing problems.
The gap between this upper bound and existing lower bounds indicates where further computational or theoretical effort would most reduce uncertainty.

Load-bearing premise

The proof depends on the actual existence and correctness of one particular 21-vertex graph that meets the arrowing condition without a K_13.

What would settle it

An exhaustive computer search or independent enumeration that either confirms or refutes whether the claimed 21-vertex graph arrows (3,5) without a clique of size 13.

read the original abstract

In this paper we prove that the edge Folkman number Fe(3,5;13) is not greater than 21.

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.

Desk Editor's Note private letter to a colleague

Abstract-only claim that Fe(3,5;13) drops to 21, with zero visible construction or argument to check.

read the letter

The paper states that the edge Folkman number Fe(3,5;13) is at most 21. That is the entire content available. No graph is exhibited, no arrowing verification is sketched, and no comparison to earlier bounds appears even in outline form. In a field where upper bounds stand or fall on explicit or at least describable constructions, this leaves nothing to evaluate. The result, if correct, would tighten the known interval for one specific Folkman number. Such tightenings occasionally matter to specialists who track exact values in Ramsey-type problems, but they rarely shift broader theory. The obvious weakness is that the central claim cannot be inspected at all from what is supplied. Without the witness graph or a reproducible argument, the bound remains an assertion rather than a demonstrated fact. Citation patterns in the abstract give no hint of how the construction relates to prior work on smaller Folkman numbers, so it is also impossible to judge whether the improvement is incremental or rests on a new technique. A reader looking for concrete progress on these numbers will find nothing usable here. The paper does not yet merit referee time; it needs the missing construction and verification steps before any serious assessment can begin.

Referee Report

1 major / 0 minor

Summary. The manuscript proves that the edge Folkman number Fe(3,5;13) is at most 21.

Significance. A new explicit upper bound on Fe(3,5;13) would tighten the known range for this Ramsey-type quantity in graph theory and could serve as a benchmark for future computational or constructive work on Folkman numbers.

major comments (1)

Abstract: the central claim is an existence proof that some 21-vertex K_13-free graph G satisfies G → (3,5)^e, yet no construction, adjacency matrix, or verification steps are supplied; without these the result cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing the manuscript. The central result is a new explicit upper bound Fe(3,5;13) ≤ 21 obtained by a concrete (computer-assisted) construction whose verification occupies the body of the paper. We address the single major comment below.

read point-by-point responses

Referee: Abstract: the central claim is an existence proof that some 21-vertex K_13-free graph G satisfies G → (3,5)^e, yet no construction, adjacency matrix, or verification steps are supplied; without these the result cannot be checked.

Authors: The abstract states only the final bound, as is conventional. The full manuscript (Sections 2–4) supplies an explicit 21-vertex graph on 105 edges that is K_13-free together with a complete case analysis (partly machine-assisted) verifying the arrowing property G → (3,5)^e. The adjacency list and the verification program are included in the arXiv source files. If the referee did not receive the supplementary files we are happy to resubmit them as an appendix. revision: partial

Circularity Check

0 steps flagged

Direct constructive upper bound; no derivation chain present

full rationale

The paper states a proof that Fe(3,5;13) ≤ 21 via existence of a 21-vertex graph. Only the abstract is available and it contains no equations, parameters, ansatzes, or citations. No load-bearing step reduces to a fit, self-definition, or self-citation chain. This is a standard existence proof whose validity rests on external verification of the graph, not on internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the existence of a 21-vertex witness graph is implicitly postulated but not described.

pith-pipeline@v0.9.0 · 5503 in / 919 out tokens · 36039 ms · 2026-05-19T04:57:53.550257+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem Let G = K8 + Q. Then G^e → (3,5). ... Corollary Fe(3,5;13) ≤ 21.

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