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arXiv · 1809.09595

A counterexample to the DeMarco-Kahn Upper Tail Conjecture

Matas Šileikis·Lutz Warnke
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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.

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Matas Šileikis, Lutz Warnke. 2018-09-25. A counterexample to the DeMarco-Kahn Upper Tail Conjecture. https://doi.org/10.1002/rsa.20859

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