Abstract
Let k and ℓ be positive integers. We prove that if 1 ≤ ℓ ≤ Ok(k6/5), then in every large enough graph G, the fraction of k-vertex subsets that induce exactly ℓ edges is at most 1/e + ok(1). Together with a recent result of Kwan, Sudakov and Tran, this settles a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn.
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Research supported by ISF grants 1028/16 and 1147/14, and ERC Starting Grant 633509 (FM), and by grant no. 200021 169242 of the Swiss National Science Foundation (MT). Part of this work has been completed at a workshop of the research group of Angelika Steger in Buchboden in July 2018.
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Martinsson, A., Mousset, F., Noever, A. et al. The edge-statistics conjecture for ℓ ≪ k6/5. Isr. J. Math. 234, 677–690 (2019). https://doi.org/10.1007/s11856-019-1929-8
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DOI: https://doi.org/10.1007/s11856-019-1929-8