Ramsey Theory

The Ramsey number (R(3,3)) is the smallest number of vertices in a graph such that any two disjoint sets of three vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(4,4)) is the smallest number of vertices in a graph such that any two disjoint sets of four vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(5,5)) is the smallest number of vertices in a graph such that any two disjoint sets of five vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(6,6)) is the smallest number of vertices in a graph such that any two disjoint sets of six vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(7,7)) is the smallest number of vertices in a graph such that any two disjoint sets of seven vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(8,8)) is the smallest number of vertices in a graph such that any two disjoint sets of eight vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(9,9)) is the smallest number of vertices in a graph such that any two disjoint sets of nine vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(10,10)) is the smallest number of vertices in a graph such that any two disjoint sets of ten vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(11,11)) is the smallest number of vertices in a graph such that any two disjoint sets of eleven vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(12,12)) is the smallest number of vertices in a graph such that any two disjoint sets of twelve vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(13,13)) is the smallest number of vertices in a graph such that any two disjoint sets of thirteen vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(14,14)) is the smallest number of vertices in a graph such that any two disjoint sets of fourteen vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(15,15)) is the smallest number of vertices in a graph such that any two disjoint sets of fifteen vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(16,16)) is the smallest number of vertices in a graph such that any two disjoint sets of sixteen vertices are either all connected by edges or all disconnected by edges.

The Ramsey number (R(17,17)) is the smallest number of vertices in a graph such that any two disjoint sets of seventeen vertices are either all connected by edges or all disconnected by edges.