A counterexample to the DeMarco-Kahn Upper Tail Conjecture

TL;DR

This paper disproves the DeMarco-Kahn Upper Tail Conjecture by providing counterexamples, revealing that the conjectured exponential decay rate for the upper tail probability in random graphs does not always hold.

Contribution

The authors construct an infinite family of graphs that violate the previously conjectured upper tail bound, challenging a longstanding assumption in random graph theory.

Findings

Counterexamples to the DeMarco-Kahn Upper Tail Conjecture

Demonstration that the conjecture does not hold universally

Insights into the behavior of upper tail probabilities in random graphs

Abstract

Given a fixed graph H, what is the (exponentially small) probability that the number X_H of copies of H in the binomial random graph G_{n,p} is at least twice its mean? Studied intensively since the mid 1990s, this so-called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of \Pr(X_H \ge (1+\epsilon) \E X_H) for fixed \epsilon>0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.