Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, September 21, 2021, 12:15 pm

Duration: 30 minutes

Location: CAB G51 and Zoom: conference room

Speaker: Rajko Nenadov (Google)

A new proof of the KŁR conjecture

We give a new, short and intuitive proof of the celebrated KŁR conjecture. Slightly rephrased, the conjecture states that if we condition on uniform edge distribution, the archetypal property of random graphs, the probability that the Erdős–Rényi random graph G(n,m) does not contain a copy of a fixed graph H becomes superexponentially small in m, for sufficiently large m > m(n, H). As its most prominent application, this conjecture implies that with high probability all subgraphs of the binomial random graph with appropriate parameters satisfy an embedding lemma which complements the sparse regularity lemma, from which many Ramsey and extremal results follow. The proof proceeds by induction and, in some way, can be considered a `deterministic' analogue of the multiple-exposure technique from random graph theory.

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