Online Talk

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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. The proof proceeds by induction and, in some way, can be considered a `deterministic' analogue of the multiple-exposure technique from random graph theory.