Abstract
Ramsey’s theorem says that if it is large enough, order must appear. This paper asks how large “large enough” is——the answer is the theorem does not say. No new mathematical theorem and no new law is claimed.
First, only two values are settled. $R(3,3)=6$ and $R(4,4)=18$; $R(5,5)$ is pinned only to the range $[43,48]$ (Section 2).
Second, this is the core of the paper. Checking $R(5,5)=43$ by exhaustion needs $2^{903}=\mathbf{10^{271.8}}$ colourings, $\mathbf{191.8}$ orders more than the atoms in the universe (Section 3).
Third, each step up jumps. $R(3,3)$ is $32768$ colourings, within reach by hand; $R(4,4)$ is $1.14\times10^{46}$, which computers barely reached (Section 3).
Fourth, the lower-bound proof does not give the value either. Erdos’s probabilistic method gives $R(k,k)>2^{k/2}$, which at $k=10$ is one $\mathbf{24.94}$th of the truth (Section 4).
Fifth, the upper bound is further off still. $W(3,3)=27$, yet the bound from the classical proof was of Ackermann type (Section 5).
Sixth, the separator is existence against quantity. A different cut from Paper 163’s existence against construction (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——Ramsey’s theorem, the known values and ranges of Ramsey numbers, van der Waerden numbers, and Erdos’s probabilistic lower bound are all standard. We do not build combinatorics——all we use is one binomial coefficient and two logarithms. We do not prove Ramsey’s theorem——we merely quote it. We do not compute Ramsey numbers——no attempt is made to find $R(5,5)$. We count only the effort it would take. We assert no values for the ranges——$R(5,5)\in[43,48]$ and $R(6,6)\in[102,160]$ are ranges known at one epoch and may be improved. The claim is that the range is not empty, not the endpoints. We do not say exhaustion is the only route——actual searches cut enormously by symmetry and pruning. $2^{903}$ is a naive upper bound, not the necessary work. The point is the order of magnitude by which it remains out of reach. We do not use “Ackermann type” strictly——it indicates that the pre-Gowers bound was of a tower-of-growing-height kind. This paper asserts no classification of the bound, only its divergence from the true value. Relation to earlier papers: Paper 163 separated “a good code exists” from “here is a good code”——that is existence against construction; this is existence against quantity. It treats the case where one can construct yet cannot reach, so the cut differs. Paper 179 showed that “cannot be constructed” has distinct roots——this paper treats what can be constructed yet is out of reach. Paper 190 measured “rare” on a logarithmic scale——the $271.8$ orders here are read logarithmically too. Paper 259 showed there are two roads to independence——this paper likewise asks what kinds of showing there are. What is added is computing the exhaustion for $R(5,5)$ as $10^{271.8}$ and measuring the gap to the atoms of the universe as $191.8$ orders, lining up the jumps at $R(3,3)$, $R(4,4)$, $R(5,5)$, computing that the probabilistic lower bound is one $24.94$th at $k=10$, and placing the separator at existence against quantity.
1. Notation and the Question
Theorem (Ramsey, $1930$): If $n$ is large enough, then colouring the edges of the complete graph on $n$ vertices with two colours, a monochromatic $K_k$ must appear.
Definition: The least such $n$ is written $R(k,k)$.
Question: The theorem says it must appear. What, then, is $R(k,k)$?
Observation 1.1 The proof guarantees existence. It does not give the value——this paper measures that gap numerically.
Observation 1.2 Erdos put it famously——if aliens demanded $R(5,5)$ humanity should commit all its computing resources, but if they demanded $R(6,6)$ we should rather fight. This paper checks that remark against numbers.
2. Only Two Values Are Settled
Check 2.1:
| Lower | Upper | Settled? | |
|---|---|---|---|
| $R(3,3)$ | $6$ | $6$ | yes |
| $R(4,4)$ | $18$ | $18$ | yes |
| $R(5,5)$ | $43$ | $48$ | no |
| $R(6,6)$ | $102$ | $160$ | no |
| $R(7,7)$ | $205$ | $492$ | no |
Observation 2.2 Only the first two are settled. $R(4,4)=18$ was settled in $1995$——sixty-five years after the theorem was proved.
Observation 2.3 The range for $R(6,6)$ is $[102,160]$, a width of $58$. For $R(7,7)$ it is $[205,492]$, a width of $287$——each step up increases the ignorance.
Observation 2.4 Yet “it exists” has been proved from the start. For every $k$, $R(k,k)$ is finite——known to be finite and unknown in value.
3. The Core: $10^{271.8}$ Colourings to Exhaust
Setting: Suppose we check $R(5,5)=43$ by exhaustion. We two-colour every edge of the complete graph on $43$ vertices.
Check 3.1:
| Quantity | Value |
|---|---|
| Vertices | $43$ |
| Edges $\binom{43}{2}$ | $\mathbf{903}$ |
| Colourings $2^{903}$ | $\mathbf{10^{271.8}}$ |
| Atoms in the observable universe | $10^{80}$ |
| Orders short | $\mathbf{191.8}$ |
Observation 3.2 This is the core of the paper. Turn every atom into a computer and one is still $\mathbf{191.8}$ orders short——what is lacking is not the speed of computers but the quantity of matter.
Check 3.3: We step down to see where reach begins.
| Vertices | Edges | Colourings | |
|---|---|---|---|
| $R(3,3)$ | $6$ | $15$ | $\mathbf{32768}$ (countable by hand) |
| $R(4,4)$ | $18$ | $153$ | $1.14\times10^{46}$ (by computer, $1995$) |
| $R(5,5)$ | $43$ | $903$ | $\mathbf{10^{271.8}}$ (out of reach) |
Observation 3.4 From $R(3,3)$ to $R(4,4)$ is $\mathbf{10^{41}}$; from $R(4,4)$ to $R(5,5)$ is $\mathbf{10^{226}}$——each step up jumps further than the last.
Observation 3.5 A point to write honestly——$2^{903}$ is a naive upper bound. Real searches cut it enormously by symmetry and pruning——only the fact that $R(5,5)$ has still not been reached grounds this paper.
4. The Lower-Bound Proof Does Not Give the Value Either
Setting: Erdos ($1947$) showed $R(k,k)>2^{k/2}$ by the probabilistic method. Colour at random; a configuration with no monochromatic $K_k$ exists with positive probability.
Check 4.1:
| $k$ | $2^{k/2}$ | Known lower bound | Ratio |
|---|---|---|---|
| $3$ | $2.83$ | $6$ | $2.12$ |
| $4$ | $4.00$ | $18$ | $4.50$ |
| $5$ | $5.66$ | $43$ | $7.60$ |
| $6$ | $8.00$ | $102$ | $12.75$ |
| $7$ | $11.31$ | $205$ | $18.12$ |
| $10$ | $32.00$ | $798$ | $\mathbf{24.94}$ |
Observation 4.2 The probabilistic bound is one $\mathbf{24.94}$th of the truth at $k=10$. And the ratio widens with $k$——it does not converge.
Observation 4.3 There is a second gap here. The method shows that a colouring with no monochromatic $K_k$ exists, and exhibits not one of them——the existence against construction of Paper 163.
Observation 4.4 So the gap is twofold.
| What is known | What is not |
|---|---|
| $R(k,k)$ is finite | its value (this paper) |
| A colouring meeting the bound exists | which colouring (Paper 163) |
Observation 4.5 The two rows are different gaps. The upper is quantity, the lower construction——one theorem carrying two kinds of “does not say” at once.
5. The Upper Bound Is Further Off Still
Setting: Van der Waerden’s theorem has the same shape——“if it is long enough, an arithmetic progression must appear.”
Check 5.1: Known values.
| Value | |
|---|---|
| $W(2,3)$ | $9$ |
| $W(3,3)$ | $\mathbf{27}$ |
| $W(4,3)$ | $76$ |
| $W(2,4)$ | $35$ |
| $W(2,5)$ | $178$ |
Observation 5.2 $W(3,3)=\mathbf{27}$. Yet the bound from the classical proof was of a tower-of-growing-height kind (Ackermann type)——the true value is $27$, and the bound could not even be written down.
Observation 5.3 Gowers ($2001$) removed the tower and brought the bound down to an elementary form. The gap to the true value remains——that a change of technique moves the bound by orders means the bound was not measuring the object.
Observation 5.4 This is the paper’s second core. A theorem’s upper bound measures the power of the proof’s tools, not the size of the object——change the tools and the bound moves; the object does not.
Observation 5.5 A point to write honestly——“Ackermann type” is used as an indication. What this paper asserts is not a classification of the bound but only that the divergence from the true value moved by orders.
6. Separator: Existence against Quantity
Observation 6.1 Separator——what must be moved to turn “it must appear” into “when it appears”? Nothing moves. They are different questions, and an answer to one does not contain the other.
| Question | Type of answer | Here |
|---|---|---|
| Does it appear? | yes/no | yes (settled $1930$) |
| At what size? | a number | open for $k\ge5$ |
| Which configuration? | a construction | not exhibited |
Observation 6.2 The three rows are answered independently. The first may be “yes” while the second and third stay open——the single word “proved” points at only one of the three.
Observation 6.3 Paper 163 separated “a good code exists” from “here is a good code.”This paper sets a third sentence beside them——“a good code lies at or below $n$.”Existence, quantity and construction are mutually independent.
Observation 6.4 Paper 190 measured “rare” on a logarithmic scale. The $\mathbf{271.8}$ orders here are the same——a size meaningless as a ratio, yet expressible as “$191.8$ orders short”.
Observation 6.5 Finally we reread Erdos’s remark numerically. $R(5,5)$ is $10^{271.8}$ colourings; $R(6,6)$ has $\binom{102}{2}=5151$ edges, hence $2^{5151}=10^{1550.6}$——a gap of $1278.8$ orders. “Rather fight” was not hyperbole but a statement of orders.
7. Roll Call and Closing
- Settled: $R(3,3)=6$, $R(4,4)=18$ (Check 2.1): $R(5,5)$ only a range.
- $R(4,4)$ settled in $1995$ (Observation 2.2): $65$ years after the theorem.
- $R(5,5)$ exhaustion is $10^{271.8}$ (Check 3.1): $903$ edges.
- $191.8$ orders past the atoms (Check 3.1): Matter, not speed, is lacking.
- One step is $10^{226}$ (Observation 3.4): The jumps grow.
- Lower bound is one $24.94$th at $k=10$ (Check 4.1): The ratio widens.
- The gap is twofold (Observation 4.4): Quantity and construction.
- $W(3,3)=27$ (Check 5.1): The bound was a tower.
- The bound measures the tools (Observation 5.4): The object does not move.
- $R(6,6)$ is $10^{1550.6}$ (Observation 6.5): $1278.8$ orders past $R(5,5)$.
Closing: Ramsey’s theorem says that if it is large enough, order must appear. But it does not say how large. Only $R(3,3)=6$ and $R(4,4)=18$ are settled, and $R(5,5)$ is pinned only to the range $[43,48]$——even settling $R(4,4)$ took $65$ years from the theorem. Counting the work of exhaustion shows why——$R(5,5)=43$ has $\mathbf{903}$ edges and $2^{903}=\mathbf{10^{271.8}}$ colourings, $\mathbf{191.8}$ orders more than the atoms in the observable universe. What is lacking is not the speed of computers but the quantity of matter. The lower bound is no better——the probabilistic method gives $R(k,k)>2^{k/2}$, one $\mathbf{24.94}$th of the truth at $k=10$, and the ratio widens with $k$. The upper bound is further off still——$W(3,3)$ is truly $\mathbf{27}$, yet the classical bound could not be written down. A theorem’s upper bound measures the power of the proof’s tools, not the size of the object. One thing separates them——whether it exists, how large it is, and which one it is, are three different questions. The first may be “yes” while the other two stay open. The single word “proved” points at only one of the three.
References
- Ramsey, F. P. (1930). On a Problem of Formal Logic. Proceedings of the London Mathematical Society, s2-30(1).
- Erdos, P. (1947). Some Remarks on the Theory of Graphs. Bulletin of the American Mathematical Society, 53(4).
- McKay, B. D. and Radziszowski, S. P. (1995). $R(4,5)=25$. Journal of Graph Theory, 19(3).
- Radziszowski, S. P. (2021). Small Ramsey Numbers. The Electronic Journal of Combinatorics, Dynamic Survey DS1.
- van der Waerden, B. L. (1927). Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wiskunde, 15.
- Gowers, W. T. (2001). A New Proof of Szemeredi’s Theorem. Geometric and Functional Analysis, 11(3).
- Alon, N. and Spencer, J. H. (2016). The Probabilistic Method. Hoboken: Wiley.
- Yamagishi, Y. (2026). The Allotted Region and the Freedom Within [Paper 112]. Zenodo. 10.5281/zenodo.21962834
- Yamagishi, Y. (2026). “A Good Code Exists” and “Here Is a Good Code” Are Different Sentences [Paper 163]. Zenodo. 10.5281/zenodo.22092677
- Yamagishi, Y. (2026). “Cannot Be Constructed” Has Distinct Roots [Paper 179]. Zenodo. 10.5281/zenodo.22098419
- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). There Are Two Roads to Independence: One Narrows the Universe and the Other Widens It [Paper 259]. Zenodo. 10.5281/zenodo.22169162
Reference
Yamagishi, Y. (2026). “It Must Appear” Does Not Tell You When [Paper 302]. YOI: https://www.yamagishiyuuki.com/yoi/302
DOI: 10.5281/zenodo.22171988
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 302. This paper is a preprint and has not been peer reviewed.