Concentration inequalities in extremal set theory

Consider a family $F$ of $k$-sets in $[n]$ of size $\alpha \binom{n}{k}$ and let $A$ be a random $k$-set. It is easy to see that on average $A$ is disjoint with $\alpha \binom{n - k}{k}$ sets of $F.$ In this work we show that the number of sets, which are disjoint with $A,$ is exponentially concentrated around its mean. We also use this to obtain some (non-probabilistic) extremal set theory results.
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000