For those who want a prefilter, Ramsey numbers are the minimum number of guests, written R(m, n), that must be invited so that at least m guests will know each other or at least n guests will not know each other.
Or in math: the minimum number of vertices in a fully connected graph that guarantee a clique of order m, or an independent set of order n.
But... That's not part of the definition of Ramsey numbers. Often it's defined in terms of a complete graph where the egdmmdges are coloured either red or blue. But there's nothing about the blue subgraph or red subgraph needing to be connected.
Sure, Ramsey's Theorem guarantees that Ramsey numbers exist. But at no point is it assumed that the graph is connected. The section you linked puts it quite simply:
> The Ramsey number is the minimum number of vertices, v = R(m, n), such that all undirected simple graphs of order v, contain a clique of order m, or an independent set of order n.
The graph is undirected and simple. But it need not be connected.
Related thread about a year ago https://news.ycombinator.com/item?id=38590156
For those who want a prefilter, Ramsey numbers are the minimum number of guests, written R(m, n), that must be invited so that at least m guests will know each other or at least n guests will not know each other.
Or in math: the minimum number of vertices in a fully connected graph that guarantee a clique of order m, or an independent set of order n.
What do you mean by "fully connected" here?
For every set of two vertices, there is at least one path that connects them.
But... That's not part of the definition of Ramsey numbers. Often it's defined in terms of a complete graph where the egdmmdges are coloured either red or blue. But there's nothing about the blue subgraph or red subgraph needing to be connected.
https://en.m.wikipedia.org/wiki/Ramsey%27s_theorem
Scroll down a bit to https://en.m.wikipedia.org/wiki/Ramsey%27s_theorem#Ramsey_nu.... Ramsey numbers arise from the original problem, and play a role in its proof, but they are not "the same thing" as Ramsey's Theorem.
Sure, Ramsey's Theorem guarantees that Ramsey numbers exist. But at no point is it assumed that the graph is connected. The section you linked puts it quite simply:
> The Ramsey number is the minimum number of vertices, v = R(m, n), such that all undirected simple graphs of order v, contain a clique of order m, or an independent set of order n.
The graph is undirected and simple. But it need not be connected.