Ting-Wei Chao, Entropy method and mixture bound

The entropy method has been used in many recent works in extremal combinatorics. With the help of Shannon entropy, significant progress has been made on several classical problems, such as the union-closed conjecture and the Sidorenko conjecture. In our recent work, we use the entropy method to give new proofs of the Kruskal–Katona theorem and Turán’s theorem, as well as some of their generalizations. The new ingredient in our approach is a method for upper bounding the sum of $2^{\mathbb{H}(X_i)}$  for random variables $X_1,\cdots,X_k$ whose supports do not overlap too much. We call this method the mixture bound, and it can be viewed as an entropic version of double counting. In this talk, I will introduce the mixture bound and show some examples of how it can be applied on colorful versions of the Kruskal–Katona theorem. Base on joint work with Maya Sankar and Hung-Hsun Hans Yu.