Ramsey Number Calculator
Use this calculator to find known Ramsey numbers R(r, s) or to understand the bounds for values that are currently unknown.
Result:
Enter values and click 'Calculate'.
function calculateRamseyNumber() {
var r = parseInt(document.getElementById("redCliqueSize").value);
var s = parseInt(document.getElementById("blueCliqueSize").value);
var resultDiv = document.getElementById("ramseyResult");
if (isNaN(r) || isNaN(s) || r < 1 || s < 1) {
resultDiv.innerHTML = "Please enter valid positive integers for r and s.";
return;
}
// Handle symmetry: R(r, s) = R(s, r)
var minVal = Math.min(r, s);
var maxVal = Math.max(r, s);
var ramseyNumbers = {
"1,1": 1, "1,2": 1, "1,3": 1, "1,4": 1, "1,5": 1, "1,6": 1, "1,7": 1, "1,8": 1, "1,9": 1, "1,10": 1,
"2,2": 2, "2,3": 3, "2,4": 4, "2,5": 5, "2,6": 6, "2,7": 7, "2,8": 8, "2,9": 9, "2,10": 10,
"3,3": 6, "3,4": 9, "3,5": 14, "3,6": 18, "3,7": 23, "3,8": 28, "3,9": 36,
"4,4": 18, "4,5": 25,
"5,5": "43-48 (bounds)" // R(5,5) is known to be between 43 and 48
};
var key = minVal + "," + maxVal;
var knownValue = ramseyNumbers[key];
if (knownValue !== undefined) {
if (typeof knownValue === 'string') {
resultDiv.innerHTML = "R(" + r + ", " + s + ") is known to be in the range: " + knownValue + ".";
} else {
resultDiv.innerHTML = "R(" + r + ", " + s + ") = " + knownValue + ".";
}
} else {
// For larger values, we can provide general bounds or state it's unknown
// General bounds: R(r,s) = 2^(min(r,s)/2)
var lowerBound = Math.pow(2, Math.min(r, s) / 2);
var upperApprox = "unknown"; // A simple upper bound is hard to calculate without R(r-1,s) and R(r,s-1)
if (r === 1 || s === 1) { // R(1,s) = 1, R(r,1) = 1
resultDiv.innerHTML = "R(" + r + ", " + s + ") = 1. (Any graph with 1 vertex guarantees a K1).";
} else if (r === 2) { // R(2,s) = s
resultDiv.innerHTML = "R(" + r + ", " + s + ") = " + s + ". (Any graph with " + s + " vertices guarantees a K2 or a K" + s + ").";
} else if (s === 2) { // R(r,2) = r
resultDiv.innerHTML = "R(" + r + ", " + s + ") = " + r + ". (Any graph with " + r + " vertices guarantees a K" + r + " or a K2).";
} else {
resultDiv.innerHTML = "R(" + r + ", " + s + ") is currently unknown or has very wide bounds. A lower bound is approximately " + Math.ceil(lowerBound) + ".";
}
}
}
.ramsey-calculator-container {
font-family: 'Arial', sans-serif;
background-color: #f9f9f9;
padding: 20px;
border-radius: 8px;
box-shadow: 0 2px 5px rgba(0,0,0,0.1);
max-width: 600px;
margin: 20px auto;
border: 1px solid #ddd;
}
.ramsey-calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 15px;
}
.ramsey-calculator-container p {
color: #555;
text-align: center;
margin-bottom: 20px;
line-height: 1.6;
}
.calculator-form .form-group {
margin-bottom: 15px;
display: flex;
align-items: center;
justify-content: space-between;
}
.calculator-form label {
flex: 1;
color: #444;
font-weight: bold;
margin-right: 10px;
}
.calculator-form input[type="number"] {
flex: 2;
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
width: calc(100% – 120px); /* Adjust width considering label */
box-sizing: border-box;
}
.calculator-form button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 16px;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
.calculator-form button:hover {
background-color: #0056b3;
}
.calculator-result {
margin-top: 25px;
padding: 15px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 5px;
text-align: center;
}
.calculator-result h3 {
color: #28a745;
margin-top: 0;
margin-bottom: 10px;
}
#ramseyResult {
font-size: 1.1em;
color: #333;
font-weight: bold;
}
Understanding Ramsey Numbers and Ramsey Theory
Ramsey Theory is a branch of mathematics that studies the conditions under which order must appear. It's often described as "complete disorder is impossible." In simpler terms, if you have a sufficiently large structure, you are guaranteed to find some substructure with a specific property, no matter how the original structure is arranged or colored.
What are Ramsey Numbers R(r, s)?
The most famous concept in Ramsey Theory is the Ramsey number, denoted as R(r, s). It represents the smallest integer 'n' such that any complete graph with 'n' vertices (K_n), whose edges are colored with two colors (say, red and blue), must contain either a red K_r (a complete subgraph with 'r' vertices, all edges red) or a blue K_s (a complete subgraph with 's' vertices, all edges blue).
Think of it this way: if you have 'n' people, and every pair of people are either friends (red edge) or strangers (blue edge), R(r, s) is the minimum number of people 'n' you need to guarantee that there are either 'r' people who are all mutual friends, or 's' people who are all mutual strangers.
The "Friends and Strangers" Problem: R(3, 3) = 6
A classic example is R(3, 3) = 6. This means that if you have 6 people at a party, there must be at least 3 people who are all mutual friends, or 3 people who are all mutual strangers. If you only have 5 people, it's possible to arrange friendships such that neither condition is met. For instance, imagine a pentagon where adjacent people are friends and non-adjacent people are strangers. No group of 3 friends or 3 strangers exists.
Properties of Ramsey Numbers:
- Symmetry: R(r, s) = R(s, r). The order of 'r' and 's' doesn't matter.
- Trivial Cases:
- R(1, s) = 1: If you want a clique of size 1 (a single vertex), you only need 1 vertex.
- R(2, s) = s: If you want a red K2 (a single red edge) or a blue K_s, you need 's' vertices. If there are 's' vertices and no blue K_s, then all edges must be red, which means you have a red K_s, which contains a red K2. If there is any red edge, you have a red K2. If there are no red edges, all edges are blue, forming a blue K_s.
- Bounds: There are general bounds for Ramsey numbers, such as R(r, s) ≤ R(r-1, s) + R(r, s-1).
The Challenge of Computing Ramsey Numbers
Despite their simple definition, Ramsey numbers are notoriously difficult to compute. Only a handful of exact Ramsey numbers are known, especially for larger values of 'r' and 's'. The difficulty arises because finding R(r, s) requires proving that a certain configuration *must* exist for 'n' vertices, and that it's *possible* to avoid it for 'n-1' vertices. This often involves extensive combinatorial searches and proofs.
For example, R(3, 3) = 6, R(3, 4) = 9, R(4, 4) = 18. However, R(5, 5) is only known to be between 43 and 48, and R(6, 6) is only known to be between 102 and 165. The exact values for many larger Ramsey numbers remain one of the most challenging open problems in combinatorics.
This calculator provides known exact values for small R(r, s) and indicates when only bounds are available or when the number is completely unknown.