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In this lecture, Sahar Diskin, a PhD student at Tel Aviv University, discusses research on long cycles in percolated hypercubes. Learn about the binary d-dimensional hypercube graph structure and how percolation affects cycle length. Discover the proof that when edges are retained with probability p > C/d, the percolated hypercube typically contains a cycle of length at least (1-ε)2^d. This result confirms a long-standing folklore conjecture and answers questions posed by Condon, Espuny Díaz, Girão, Kühn, and Osthus. The presentation draws parallels to classical results by Ajtai, Komlós, Szemerédi, and Fernandez de la Vega regarding cycle lengths in random graphs. This joint work with Michael Anastos, Joshua Erde, Mihyun Kang, Michael Krivelevich and Lyuben Lichev represents significant advancement in probabilistic combinatorics and graph percolation theory.
