# Tópicos em Combinatória Extremal

In this advanced-level course we will study some recent developments in extremal combinatorics, including:

• The absorption method: applications to decompositions of graphs and hypergraphs.
• Recent developments in graph Ramsey theory.
• The proof of the Erdös–Faber–Lovász conjecture by Kang, Kelly, Kühn, Methuku and Osthus.
• The proof of the Sensitivity Conjecture by Hao Huang.
• The resolution of the Happy Ending Problem by Andrew Suk.
• The breakthrough of Alweiss, Lovett, Wu and Zhang on the Sunflower Conjecture of Erdös and Rado.
• Recent developments on Turán numbers of bipartite graphs.
• The sparse blow-up lemma of Allen, Böttcher, Hán, Kohayakawa and Person.
• Applications of the asymmetric and efficient hypergraph container lemmas.
• Discrepancy: Littlewood polynomials, the Beck–Fiala theorem.
• Applications of the polynomial method in combinatorics, including the solution of the cap-set problem by Croot, Lev and Pach.
• Recent advances on sphere packing in high dimensions.
• Applications of combinatorial techniques in number theory.

References:
[1] R. Alweiss, S. Lovett, K. Wu and J. Zhang, Improved bounds for the sunflower lemma, Ann. Math., 194 (2021), 795–815.
[2] B. Bollobás, Modern Graph Theory (Graduate Texts in Mathematics), Springer-Verlag, New York, 1998.
[3] H. Cohn and Y. Zhao, Sphere packing bounds via spherical codes, Duke Math. J., 163 (2014), 1965–2002.
[4] S. Glock, D. Kühn and D. Osthus, Extremal aspects of graph and hypergraph decomposition problems, Surveys in Combinatorics, Cambridge University Press, 2021.
[5] L. Guth, Polynomial Methods in Combinatorics, American Mathematical Society, 2016.
[6] H. Huang, Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture, Ann. Math., 190 (2019), 949–955.
[7] A. Suk, On the Erdos-Szekeres convex polygon problem, J. Amer. Math. Soc., 30 (2017), 1047–1053.
[8] D. Conlon, J. Fox and B. Sudakov, Recent developments in graph Ramsey theory, Surveys in Combinatorics, Cambridge University Press, 2015.
[9] C. Lee, Ramsey numbers of degenerate graphs, Ann. Math., 185 (2017), 791-829.
[10] A. Sah, Diagonal Ramsey via effective quasirandomness, Duke Math. J., to appear.