Stability results for random discrete structures

Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long-standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. The theorem of Schacht can be applied in a more general setting and yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erdos and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting that is somewhat more general and stronger than the one obtained by Conlon and Gowers. We then use this theorem to derive several new results, among them a random version of the Erdos-Simonovits stability theorem for arbitrary graphs. The main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest.Comment: 19 pages; slightly simplified proof of the main theorem; fixed a few typo

Counting substructures III: quadruple systems

For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Furedi, Keevash, Pikhurko, Simonovits and Sudakov who proved under the same conditions that there is one copy of $F$. Our proofs use the hypergraph removal Lemma and stability results for the corresponding Turan problem proved by the above authors

Embedding graphs into larger graphs: results, methods, and problems

Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which are new breakthrough results, or which --- for some other reasons --- are very close to us. Some results discussed here got stronger emphasis, since they are connected to Lov\'asz (and sometimes to us).Comment: 153 pages, 15 figures, 3 tables. Almost final version of the survey for Building Bridges II (B\'ar\'any et al eds.), Laszlo Lovasz 70th Birthda

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4-books of three pages. (English summary

Previous Up Next Article Citations From References: 2 From Reviews: 0

4-books of three pages. (English summary

3

4-books of three pages Zolt'an F"uredi,