The previous talk in the series is here. The next talk in the series is here.
In the previous video I stated and proved Shearer’s entropy lemma. Here I give two applications. The first provides an upper bound for the number of triangles a graph with m edges can have. The second is an upper bound for the size of a family of graphs with vertex set {1,2,…,n} if the intersection of any two graphs in the family contains a triangle. We start with a reminder of Shearer’s lemma and give a bound for the number of triangles, then the bound for the size of a triangle-intersecting family of graphs.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.