On the Second Kahn–Kalai Conjecture
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For any given graph H, we are interested in p_crit(H), the minimal p such that the Erdős-Rényi graph G(n,p) contains a copy of H with probability at least 1/2. Kahn and Kalai (2007) conjectured that p_crit(H) is given up to a logarithmic factor by a simpler "subgraph expectation threshold" p_E(H), which is the minimal p such that for every subgraph H'⊆ H, the Erdős-Rényi graph G(n,p) contains in expectation at least 1/2 copies of H'. It is trivial that p_E(H) ≤ p_crit(H), and the so-called "second Kahn-Kalai conjecture" states that p_crit(H) ≲ p_E(H) log e(H) where e(H) is the number of edges in H. In this article, we present a natural modification p_E, new(H) of the Kahn–Kalai subgraph expectation threshold, which we show is sandwiched between p_E(H) and p_crit(H). The new definition p_E, new(H) is based on the simple observation that if G(n,p) contains a copy of H and H contains many copies of H', then G(n,p) must also contain many copies of H'. We then show that p_crit(H) ≲ p_E, new(H) log e(H), thus proving a modification of the second Kahn–Kalai conjecture. The bound follows by a direct application of the set-theoretic "spread" property, which led to recent breakthroughs in the sunflower conjecture by Alweiss, Lovett, Wu and Zhang and the first fractional Kahn–Kalai conjecture by Frankston, Kahn, Narayanan and Park.
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