# Disproof of a packing conjecture of Alon and Spencer

## TL;DR

This paper disproves a 1992 conjecture by Alon and Spencer, showing that a typical random graph does not admit a large edge coverage by nearly maximum cliques, using probabilistic combinatorics and independence heuristics.

## Contribution

It provides the first disproof of the conjecture and offers insights into the intersection properties of random subsets, challenging previous heuristic assumptions.

## Key findings

- The conjecture that random graphs can be covered by nearly maximum cliques is false.
- Analysis of intersection probabilities of random subsets reveals limitations of independence heuristics.
- The disproof relies on understanding the intersection structure of random k-subsets and their matchings.

## Abstract

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph $G_{n,1/2}$ typically admits a covering of a constant fraction of its edges by edge-disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: for $k\ll \sqrt{n}$ and $A_1\dots A_t$ chosen uniformly and independently from the $k$-subsets of $\{1\dots n\}$, what can one say about \[ \mathbb{P}(|A_i\cap A_j|\leq 1 ~\forall i\neq j)? \] Our main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.