Ramsey Number Estimator
Ramsey Number Estimator estimates Ramsey numbers \\( R(s, t) \\), the smallest number of vertices \\( n \\) such that any two-coloring (red/blue) of the edges of a complete graph \\( K_n \\) contains a red clique of size \\( s \\) or a blue clique of size \\( t \\). Enter positive integers \\( s \\) and \\( t \\) (e.g., 3, 3). The calculator provides known values or bounds using combinatorial methods, shows steps with MathJax, and visualizes a sample graph coloring with p5.js.
Ramsey Number Estimator
This calculator estimates Ramsey numbers \\( R(s, t) \\), the smallest number of vertices \\( n \\) such that any two-coloring of the edges of a complete graph \\( K_n \\) contains either a red clique of size \\( s \\) or a blue clique of size \\( t \\). Input positive integers \\( s \\) and \\( t \\). For known values (e.g., \\( R(3,3)=6 \\)), the exact number is provided; otherwise, bounds are computed using combinatorial and probabilistic methods. Steps are shown with MathJax, and a sample graph coloring is visualized with p5.js. Note: Exact Ramsey numbers are known only for small cases due to computational complexity.
Example 1: Party Problem
Input: \\( s=3 \\), \\( t=3 \\).
Result: \\( R(3,3)=6 \\).
Meaning: Any two-coloring of \\( K_6 \\) has a monochromatic triangle.
Example 2: Larger Cliques
Input: \\( s=4 \\), \\( t=3 \\).
Result: \\( R(4,3)=9 \\).
Meaning: Any two-coloring of \\( K_9 \\) has a red \\( K_4 \\) or blue \\( K_3 \\).
Example 3: Unknown Case
Input: \\( s=5 \\), \\( t=5 \\).
Result: \\( 43 \leq R(5,5) \leq 48 \\).
Meaning: Bounds for \\( R(5,5) \\) are known, but the exact value is not.