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Small Ramsey Numbers

Actually determining the Ramsey numbersR(m,n)Ramsey number R(m,n) referenced in seems to be a notoriously difficult problem, and only a handful of these values are known precisely.

Small Ramsey Numbers

Actually determining the Ramsey numbers\(R(m,n)\)Ramsey number \(R(m,n)\) referenced in seems to be a notoriously difficult problem, and only a handful of these values are known precisely. In particular, \(R(3,3)=6\) and \(R(4,4)=18\), while \(43\le R(5,5)\le 49\). The distinguished Hungarian mathematician Paul Erdős said on many occasions that it might be possible to determine \(R(5,5)\) exactly, if all the world's mathematical talent were to be focused on the problem. But he also said that finding the exact value of \(R(6,6)\) might be beyond our collective abilities.

In the following table, we provide information about the Ramsey numbers \(R(m,n)\) when \(m\) and \(n\) are at least \(3\) and at most \(9\). When a cell contains a single number, that is the precise answer. When there are two numbers, they represent lower and upper bounds.

For additional (or more current) data, see Dynamic Survey #DS1: Small Ramsey Numbers by Stanisław Radziszowski in the Electronic Journal of Combinatorics. ( was last updated using the 12 January 2014 version of that article.)

Symbols used here

\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\bar{x},\ \mu
sample mean, population mean
Average of the data; average of the whole population.
\sigma,\ s,\ \sigma^2
standard deviation, sample s.d., variance
Typical distance from the mean; its square.
P(A),\ P(A \mid B)
probability, conditional probability
Chance of A; chance of A given that B happened.
E[X],\ \operatorname{Var}(X)
expected value, variance
Probability-weighted average of X; its spread.
N(\mu, \sigma^2),\ z
normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.
\mu(A),\ \sigma\text{-algebra}
measure of A
Size of a set; the family of sets that can be measured.

Questions people ask

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

Kokeile omaasi

Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

Lisää Probability Theory

Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem