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ERDőS #627 · PARTIAL

Erdős problem #627 — live audit, exact finite values, and the remaining wall

Date of audit and computation: 2026-07-27 UTC.

Outcome

The asymptotic limit remains open. I obtained and independently checked the following exact finite result:

\[ f(n)= \begin{cases} 1,&1\leq n\leq4,\\[2mm] \frac32,&5\leq n\leq10,\\[2mm] 2,&11\leq n\leq16. \end{cases} \]

The values through \(n=12\) have a fresh exhaustive computational certificate. The extension from \(12\) through \(16\) is a short deduction from Gallai's critical-graph theorem and two published exact Folkman-number computations; the two order-\(16\) exceptional graphs are independently rechecked by the supplied script.

Claim labels used below:

Step 0: mandatory live-page audit

I accessed both the live problem page and its discussion thread through the Bright Data browser path, not direct datacenter curl.

Verbatim current statement

> Let \(\omega(G)\) denote the clique number of \(G\) and \(\chi(G)\) the chromatic number. If \(f(n)\) is the maximum value of \(\chi(G)/\omega(G)\), as \(G\) ranges over all graphs on \(n\) vertices, then does\[\lim_{n\to\infty}\frac{f(n)}{n/(\log_2n)^2}\]exist?

Source: live problem page and its live LaTeX view.

Live status and collaboration markers

The page was last edited 8 February 2026 and displayed:

Therefore the mandatory stop condition did not fire.

Results listed on the live page

The page records:

1. Tutte and Zykov independently constructed, for every \(k\), a graph with \(\omega=2\) and \(\chi=k\).

2. Erdős [Er61d] obtained \(n\)-vertex graphs with \(\omega=2\) and \(\chi\gg n^{1/2}/\log n\).

3. Erdős [Er67c] proved

\[ f(n)\asymp \frac{n}{(\log_2 n)^2}, \]

with any possible normalized limit lying in \([1/4,4]\). The page explains that Erdős printed \(1\) as the upper endpoint, whereas Araujo–Filipe–Miyazaki observe that the method gives \(4\).

4. Araujo–Filipe–Miyazaki [AFM25] prove that, if the diagonal Ramsey exponential rate

\[ \lim_{k\to\infty}\frac{\log_2 R(k,k)}{k}=C \]

exists and \(R(s,t)\leq R(k,k)\) whenever \(st\leq k^2\), then the limit in #627 exists and equals \(C^2\). The page also records their improved upper constant, approximately \(3.7\).

The one live comment

The discussion thread contained one comment, by Adenwalla on 19 December 2025. It pointed to the Araujo–Filipe–Miyazaki preprint, its correction of the old upper constant \(1\) to \(4\), and its new constant \(c<3.72\). The comment itself says that the site was updated to incorporate it. It is not a proof claim, and the site labels user comments as unverified.

One bibliographic typo is visible in the live LaTeX view: it prints arXiv:252.16062. The verified current identifier is arXiv:2512.16062.

Primary-source literature check

The original problem

I inspected the scanned primary article:

The scan states the two-sided order \(n/(\log n)^2\), asks for the limit, and gives the original constants in its chosen logarithm normalization. The later 1969 survey also repeats that existence was considered likely but unproved.

The current improvement

The actual current preprint is:

Its Theorem 1.3 gives the unconditional bound

\[ f(n)\leq(3.71943+o(1))\frac{n}{(\log_2 n)^2}, \]

and, under its Conjecture 1.1, the conditional constant \(3.70831\). Its Theorems 2.1 and 2.2 identify the lower and upper asymptotic quantities in terms of diagonal and off-diagonal Ramsey exponential rates. Under Conjecture 1.1 and existence of the diagonal rate, these coincide and give the squared constant claimed on the tracker.

Exact-title, exact-constant, arXiv, and problem-number searches through 2026-07-27 found no later primary paper directly improving or resolving this limit. This is an honest search result, not a proof that no uncatalogued result exists.

New finite theorem

The table

| \(n\) | exact \(f(n)\) | source of upper bound | lower witness | label |

|---:|---:|---|---|---|

| \(1\)–\(4\) | \(1\) | fresh critical-candidate census | any complete graph, padded if needed | (d) |

| \(5\)–\(10\) | \(3/2\) | fresh critical-candidate census | \(C_5\) plus isolated vertices | (d) |

| \(11\)–\(12\) | \(2\) | fresh critical-candidate census | \(M(C_5)\) plus isolated vertices | (d) |

| \(13\)–\(16\) | \(2\) | Gallai + Jensen–Royle + Lathrop–Radziszowski | \(M(C_5)\) plus isolated vertices | (b) |

Here \(M(C_5)\) is the 11-vertex Mycielskian (the Grötzsch graph). The checker constructs it rather than loading a canned graph and obtains

\[ |V|=11,\qquad |E|=20,\qquad \chi=4,\qquad\omega=2. \]

Thus it has ratio \(2\). It also reconstructs \(C_5\) and obtains \((|V|,|E|,\chi,\omega)=(5,5,3,2)\). Adding isolated vertices changes neither \(\chi\) nor \(\omega\). (a) for the construction, (d) for the independent machine check.

Why the finite census is exhaustive

The following reduction is elementary. (a)

Fix a rational target \(\rho=p/q\). Suppose \(G\) satisfies

\[ \frac{\chi(G)}{\omega(G)}>\rho. \]

Choose an induced subgraph \(H\) minimal by vertex count among induced subgraphs with this property. Put \(k=\chi(H)\) and \(w=\omega(H)\).

1. \(H\) is vertex-critical. If \(\chi(H-v)=k\), then \(\omega(H-v)\leq w\), so \(H-v\) still violates the target, contradicting minimality.

2. \(H\) is connected. A vertex-critical graph with chromatic number at least \(2\) cannot have a second component: deleting a vertex in a component not carrying the maximum chromatic number would leave \(\chi\) unchanged.

3. \(\delta(H)\geq k-1\). Otherwise, color \(H-v\) with \(k-1\) colors and give \(v\) a color missing from its at most \(k-2\) neighbors.

4. Since \(qk>pw\),

\[ k\geq K(w):=\left\lfloor\frac{pw}{q}\right\rfloor+1. \]

Therefore \(H\) is connected, \(K_{w+1}\)-free, and has minimum degree at least \(K(w)-1\).

Consequently it is enough, for every feasible \(w\) and every order \(K(w)\leq m\leq N\), to enumerate connected \(m\)-vertex graphs with minimum degree at least \(K(w)-1\), discard those containing \(K_{w+1}\), and test \((K(w)-1)\)-colorability. Every possible violator appears in this search, up to isomorphism. The case \(w=1\) is empty because a nonempty graph with clique number \(1\) has chromatic number \(1\).

Computational certificate through \(n=12\)

The standalone checker is erdos627_wave6g_reverify.py.

Run from the repository root:

python3 runs/erdos627_wave6g_reverify.py

Environment used:

The script:

1. parses graph6 itself and self-tests the parser on all four unlabeled 3-vertex graphs;

2. invokes geng only as the exhaustive nonisomorphic graph generator;

3. independently rechecks every generated graph's order, minimum degree, and forbidden clique condition;

4. runs two separately implemented exact colorability algorithms on every clique-eligible graph:

5. aborts on any disagreement or any candidate violator;

6. constructs and exactly checks \(C_5\), \(M(C_5)\), and the two complete order-16 Ramsey \((4,4)\) data records.

The final full run returned PASS in 10.905 seconds:

| target excluded | generator rows | generated (duplicates between rows allowed) | clique-eligible | noncolorable / violators |

|---|---:|---:|---:|---:|

| \(\chi/\omega>1\), order \(\leq4\) | 3 | 1 | 1 | 0 |

| \(\chi/\omega>3/2\), order \(\leq10\) | 21 | 71,384 | 68,267 | 0 |

| \(\chi/\omega>2\), order \(\leq12\) | 20 | 69,439 | 33,043 | 0 |

Three large reproducibility rows are:

| target | \(w\) | order | degree floor | generated | eligible | SHA-256 of graph6 stream |

|---|---:|---:|---:|---:|---:|---|

| \(>3/2\) | 3 | 10 | 4 | 66,107 | 66,107 | d94337c59b42819151b6c6aed4d574643597e633096fc53f4e9ab3a618b50a8d |

| \(>2\) | 3 | 12 | 6 | 32,418 | 32,418 | 71559c4977b160d8cf2f5eda3632c88b2fc0c8e0bf6046f2dc104a3555fdcf8c |

| \(>2\) | 4 | 12 | 8 | 36,327 | 100 | c0ab653e4f1bbf62db528916b37ab391770602208c16436be17049b28cd9650c |

The full output prints analogous counts and hashes for every row. No stored table is trusted by the program; each invocation regenerates and retests the graphs.

Current file hashes:

d11daa8a45e55763fd7adb96b9b7fc3021342a480fcc6737b745a50ab04e9506  runs/erdos627_wave6g_reverify.py
9730b53764bdb28ecd2fdf755fafbc76992050f39e5ea19bb7d91433a26583e9  /usr/bin/nauty-geng

The theorem-assisted extension to \(n=16\)

Assume a graph of order at most \(16\) has \(\chi>2\omega\), and choose a vertex-minimal violator \(H\) as above. (a) The complement \(\overline H\) must be connected: otherwise \(H\) is the join of the induced graphs \(H_1,\ldots,H_t\) on the components of \(\overline H\), and

\[ \chi(H)=\sum_i\chi(H_i),\qquad \omega(H)=\sum_i\omega(H_i). \]

If the first sum is more than twice the second, some \(H_i\) is a smaller violator.

Gallai's theorem says that a \(k\)-critical graph with connected complement has at least \(2k-1\) vertices. Therefore (b)

\[ |V(H)|\geq2\chi(H)-1\geq2(2\omega(H)+1)-1=4\omega(H)+1. \]

If \(|V(H)|\leq16\), this forces \(\omega(H)\leq3\).

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  Ov?IXZIhlRWjUXL[iphst

The standalone checker independently obtains \((|V|,|E|,\chi,\omega,\alpha)=(16,60,6,3,3)\) for both. Thus neither has \(\chi\geq7\). (b) for completeness of the published classification, (d) for the invariant recheck.

This excludes every ratio strictly greater than \(2\) through order \(16\). The padded \(M(C_5)\) gives equality.

Primary sources used here:

Exact finite frontier at \(n=17\)

The same minimal-counterexample argument shows exactly what first remains at order \(17\). A ratio \(>2\) can only arise from one of:

1. a \(K_4\)-free graph with \(\chi\geq7\), equivalently the order-17 case of \(F_v(2^6;4)\);

2. a \(K_5\)-free graph with \(\chi\geq9\), equivalently the order-17 case of \(F_v(2^8;5)\).

The \(\omega=2\) branch is excluded below order \(22\) by Jensen–Royle, and Gallai excludes \(\omega\geq5\) because \(4\omega+1\geq21\). Thus this two-branch statement is an exact reduction, not a heuristic. (a)+(b)

A naive geng route is already unattractive before order 17. A measured \(1/100\) residue of the \(K_4\)-free, minimum-degree-6 order-13 generation contained 155,111 graphs and took 3.05 CPU-seconds. The full order-12 row had 32,418 graphs. Extrapolating even a conservative factor \(100\) per added vertex from the order-13 sample would put a naive order-17 scan near \(10^{15}\) candidates; at the observed approximately \(8\times10^3\) independently double-checked candidates/second, that is about \(5\times10^7\) core-hours, or roughly USD 2.5 million at USD 0.05 per core-hour. This extrapolation is (c), not a lower bound. A serious order-17 attack would require a specialized maximal-Folkman extension or proof-producing SAT formulation; I did not launch such a computation.

Why this does not settle the asymptotic limit

The exact values above concern a fixed finite range and provide no uniform control as \(n\to\infty\). The clean current asymptotic reduction is the one in Araujo–Filipe–Miyazaki. Two missing ingredients remain:

1. existence of the diagonal Ramsey exponential rate

\[ \lim_{k\to\infty}\frac{\log_2R(k,k)}{k}; \]

2. their multiplicative diagonal domination conjecture

\[ st\leq k^2\quad\Longrightarrow\quad R(s,t)\leq R(k,k). \]

Their Appendix notes that even the much narrower comparison \(R(t-1,t+1)\leq R(t,t)\) has no relevant progress. Thus standard Ramsey bounds currently control the liminf and limsup constants but do not identify them. This is the exact missing lemma/uniformity step; neither the finite census nor known random-graph estimates bridge it.

PARTIAL: Exact, reproducibly checked values are \(f(n)=1\) for \(1\le n\le4\), \(f(n)=3/2\) for \(5\le n\le10\), and \(f(n)=2\) for \(11\le n\le16\); the asymptotic limit remains open.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger