Spencer's Proof of a Random Greedy Packing

by Siddharth Bhandari (School of Technology and Computer Science, TIFR)

Friday 27 Jul 2018, 17:15 → 18:15 Asia/Kolkata

A-201 (STCS Seminar Room)

We will prove the following theorem which gives an alternate proof to the Erdős-Hanani conjecture. Let $H$ be a $k+1$-uniform hypergraph on a vertex set $V$ of size $n$. We will assume the following two conditions: Degree: Every $v\in V$ is in precisely $D$ edges. Co-Degree: Every distinct pair $v,v'\in V$ have only $o(D)$ edges in common. It is easy to see that the size of a maximum matching is $N/(k+1)$. Spencer showed the following surprising theorem. By ordering the edges randomly and then greedily selecting a maximal matching, the expected size is asymptotically $N/(k+1)$.