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Tuesday, April 27, 2021

Ramsey Theory and Commercial Cliques

A colleague of mine and I brought this up and how we might potentially use it to understand some kinds of consumer group commercial behavior.   Glad to be reminded of this, and  thinking of how it might be tested with more data.  Ultimately this is technically deep,  but I think does have some useful insight.

 New Proof Reveals That Graphs With No Pentagons Are Fundamentally Different  By Steve Nadis in Quanta

Researchers have proved a special case of the ErdÅ‘s-Hajnal conjecture, which shows what happens in graphs that exclude anything resembling a pentagon. 

When you walk into a room full of people, you can speculate about all sorts of things, from political leanings to TV viewing habits. But if the room has at least six people, you can say something about them with absolute mathematical certainty, thanks to a 1930 theorem by Frank Ramsey: Among those people, there’s either a group of three who all know each other, or a group of three who have never met.

The scope of Ramsey theory, which examines the patterns that emerge as a group gets larger, extends well beyond social gatherings. It also has direct and crucial implications for a branch of mathematics known as graph theory. These graphs consist of collections of points, or vertices, that may (or may not) be connected to each other by an edge — equivalent to people at a party who may (or may not) have met before. The size of a graph is set by n, the number of vertices it has. A portion of a graph in which every vertex is connected by an edge to every other vertex is, fittingly, called a clique. Conversely, a portion of a graph in which no vertex is connected to any other vertex is called an anticlique, or stable set.  ... " 

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