Edge-statistics on large graphs

A Completion of the Proof of the Edge-statistics Conjecture

“…The following observations of Alon et al [1] motivate this conjecture and show that the constant 1/e in the conjecture is the smallest possible: By considering a random graph G(n, p) with p = 1/ k 2 , one can see that ind(k, 1) ≥ e −1 + o k (1). Furthermore, by considering a complete bipartite graph with parts of sizes n/k and (k − 1)n/k, one obtains ind(k, k − 1) ≥ e −1 + o k (1).…”

“…Hence, when studying the edge-inducibility ind(k, ℓ), one may assume that ℓ ≤ k 2 /2. As mentioned above, Alon et al [1] proved that ind(k, ℓ) < 1 − ε for all 0 < ℓ < k 2 with some absolute constant ε. They furthermore established Conjecture 1.1 for Ω(k 2 ) ≤ ℓ ≤ k 2 /2 by proving that ind(k, ℓ) ≤ O(k −0.1 ) in this range.…”

“…Given integers k ≥ 1 and 0 ≤ ℓ ≤ k 2 , Alon, Hefetz, Krivelevich and Tyomkyn [1] asked the following question: If we choose a k-vertex subset uniformly at random from a very large graph, what is the maximum possible probability to obtain exactly ℓ edges within the random k-vertex subset? More precisely, for n ≥ k they defined I(n, k, ℓ) to be the maximum possible probability of the event to have exactly ℓ edges within a uniformly random k-vertex subset sampled from an n-vertex graph G (where the maximum is taken over all n-vertex graphs G).…”

“…If ℓ = 0 or ℓ = k 2 , it is easy to see that ind(k, ℓ) = 1 by taking G to be an empty or a complete graph, respectively. However, Alon et al [1] showed that for 0 < ℓ < k 2 the edge-inducibility ind(k, ℓ) is bounded away from 1. Furthermore, they formulated the following stronger conjecture, which they called the Edge-statistics Conjecture:…”

“…Conjecture 1.1 (The Edge-statistics Conjecture, [1]). For all integers k and ℓ with 0 < ℓ < k 2 , we have ind(k, ℓ) ≤ 1 e + o k (1).…”