Arsenii Sagdeev: Cutting corners

Abstract: We will discuss one simple argument leading to the following:
1) lower bound on the minimum number of colors needed to color R^n such that no unit equilateral triangle is monochromatic;
2) upper bounds on the maximum number of subsets of [n] no k of which form a weak-sunflower;
3) (perhaps the shortest) proof of the Frankl-Rődl theorem that states that the size of every family of r-element subsets of [n] no 2 of which share exactly s elements is 'small'.