A counterexample to the DeMarco‐Kahn upper tail conjecture

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  for fixed ε > 0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.