For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold". Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures, with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. M. Talagrand introduced the notion of "p-smallness" as an explicit certificate to show the p-biased product measure of a given increasing family F is small. In this talk, we will introduce various problems related to "p-smallness" of increasing families.
Based on joint works with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

Video Recording