Combinatorics Seminar

When: Sunday, December 18, 10am Where: Schreiber 309 Speaker: Po-Shen Loh, Carnegie Mellon University Title: On a problem of Erdos and Rothschild on edges in triangles

Abstract:

Erdos and Rothschild asked to estimate the maximum number, denoted by h(n,c), such that every n-vertex graph with at least cn^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least h(n,c) triangles. In particular, Erdos asked in 1987 to determine whether for every c>0 there is epsilon>0 such that h(n,c)>n^{epsilon} for all sufficiently large n. We prove that h(n,c)=n^{O(1/\log \log n)} for every fixed c<1/4. This gives a negative answer to the question of Erdos, and is best possible in terms of the range for c, as it is known that every n-vertex graph with more than n^2/4 edges contains an edge that is in at least n/6 triangles.

Joint work with Jacob Fox.