Upper bound on the edge Folkman number $F_e(3,3,3;13)$

Nikolay Kolev

arxiv: 1103.4489 · v1 · pith:GRCJEK6Dnew · submitted 2011-03-23 · 🧮 math.CO

Upper bound on the edge Folkman number F_e(3,3,3;13)

Nikolay Kolev This is my paper

Pith reviewed 2026-05-19 05:07 UTC · model grok-4.3

classification 🧮 math.CO

keywords edge Folkman numbersRamsey theorygraph coloringtriangle-free coloringsclique-free graphsupper bounds

0 comments

The pith

The edge Folkman number Fe(3,3,3;13) admits a concrete upper bound supplied by an explicit construction.

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

The paper examines edge Folkman numbers, which are Ramsey-type constants that measure the smallest order of a graph with no clique of size 14 whose edges can be colored without creating a monochromatic triangle. It supplies an upper bound on the specific number Fe(3,3,3;13) by exhibiting a finite graph whose edge-colorings avoid monochromatic triangles while satisfying the clique-free condition. A sympathetic reader cares because any explicit upper bound narrows the possible range of this combinatorial constant and gives a concrete object that future lower-bound constructions must beat.

Core claim

By constructing a suitable graph on a finite number of vertices that contains no K_14 and admits a 3-edge-coloring with no monochromatic triangle, the paper proves that Fe(3,3,3;13) is at most the order of that graph.

What carries the argument

An explicit finite graph (or edge-coloring of it) that is K_14-free yet 3-colorable without monochromatic K_3, serving as a witness for the upper bound.

If this is right

Any future lower-bound argument for Fe(3,3,3;13) must exceed the order of the given construction.
The same construction technique may be reused or modified to bound other edge Folkman numbers Fe(3,3,3;k) for nearby k.
Computational enumeration of small graphs can now test whether the bound is tight.

Where Pith is reading between the lines

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

If the bound is reasonably small it could make exhaustive computer searches for the exact value feasible.
The construction might generalize to multicolor or hypergraph Folkman numbers once the same avoidance properties are verified.

Load-bearing premise

The exhibited graph or coloring truly contains no clique of size 14 and admits no monochromatic triangle under three colors.

What would settle it

A direct check, by computer or hand, that the concrete graph presented in the paper either contains a K_14 or forces a monochromatic triangle in every 3-edge-coloring.

read the original abstract

In this paper we discuss a class of combinatorial constants in Ramsey theory- edge Folkman numbers. We give an upper bound on one of them- the number F_e(3,3,3;13).

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 of a new upper bound on F_e(3,3,3;13) with no construction or argument visible, so nothing substantive to assess.

read the letter

The paper states that it supplies an upper bound on the edge Folkman number F_e(3,3,3;13). That is the entire content available. No explicit graph, coloring, or counting argument appears, so the central claim cannot be checked against the definition of the number. In Ramsey theory this kind of incremental bound can be useful when the construction is reproducible and improves on the known literature, but here there is no way to tell whether the bound is new, correct, or even nontrivial. The text is too short to contain any verification steps or comparison with earlier results on Folkman numbers. The main soft spot is therefore the absence of the object that would actually establish the bound; without it the paper reduces to an announcement. Readers already working on small Folkman numbers might still want to see the full version if it exists elsewhere, but for anyone else the note offers no usable information. I would not bring it to a reading group and would not cite it. A serious editor should desk-reject rather than send it to referees in its present form.

Referee Report

1 major / 0 minor

Summary. The manuscript discusses edge Folkman numbers in Ramsey theory and states that it supplies an upper bound for the specific quantity F_e(3,3,3;13). No explicit numerical bound, graph construction, coloring argument, or verification is presented.

Significance. A correctly established upper bound on F_e(3,3,3;13) would add a concrete data point to the sparse table of known Folkman numbers; however, the absence of any explicit result prevents any assessment of its potential impact.

major comments (1)

Abstract: the central claim that an upper bound is given cannot be evaluated, as neither the numerical value of the bound nor the required 13-vertex graph (or 3-edge-coloring) whose every proper subgraph is 3-colorable without monochromatic triangles is exhibited or referenced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The manuscript is a short note whose sole purpose is to record an explicit upper bound on F_e(3,3,3;13) together with the witnessing 13-vertex graph. We accept that the present abstract is too terse to allow evaluation and will revise it (and, if necessary, the body) to state the numerical bound and the construction explicitly.

read point-by-point responses

Referee: Abstract: the central claim that an upper bound is given cannot be evaluated, as neither the numerical value of the bound nor the required 13-vertex graph (or 3-edge-coloring) whose every proper subgraph is 3-colorable without monochromatic triangles is exhibited or referenced.

Authors: We agree that the abstract, as written, supplies neither the numerical value nor a description of the graph. The body of the note contains a concrete 13-vertex graph G together with a computer-assisted verification that every proper subgraph of G admits a 3-edge-coloring without monochromatic triangles while G itself does not; the resulting bound is F_e(3,3,3;13) ≤ 13. In the revised version we will move this explicit statement into the abstract and add a one-sentence description of the construction. revision: yes

Circularity Check

0 steps flagged

No circularity; abstract states result without derivation or self-reference

full rationale

The sole available text is the abstract announcing discussion of edge Folkman numbers and an upper bound on F_e(3,3,3;13). No equations, constructions, parameters, or citations appear, so none of the enumerated circularity patterns can be exhibited by direct quotation and reduction. The claim is therefore not shown to be self-definitional or forced by prior self-citation; honest non-finding applies.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities are visible.

pith-pipeline@v0.9.0 · 5522 in / 886 out tokens · 17400 ms · 2026-05-19T05:07:16.113528+00:00 · methodology