Title:

Analytic methods in combinatorics

In the thesis, we apply the methods from the recently emerged theory of limits of discrete structures to problems in extremal combinatorics. The main tool we use is the framework of ag algebras developed by Razborov. We determine the minimum threshold d that guarantees a 3uniform hypergraph to contain four vertices which span at least three edges, if every linearsize subhypergraph of the hypergraph has density more than d. We prove that the threshold value d is equal to 1=4. The extremal configuration corresponds to the set of cyclically oriented triangles in a random orientation of a complete graph. This answers a question raised by Erdos. We also use the ag algebra framework to answer two questions from the extremal theory of permutations. We show that the minimum density of monotone subsequences of length �ve in any permutation is asymptotically equal to 1=256, and that the minimum density of monotone subsequences of length six is asymptotically equal to 1=3125. Furthermore, we characterize the set of (su�ciently large) extremal con�gurations for these two problems. Both the values and the characterizations of extremal con�gurations were conjectured by Myers. Flag algebras are also closely related to the theory of dense graph limits, where the main objects of study are convergent sequences of graphs. Such a sequence can be assigned an analytic object called a graphon. In this thesis, we focus on finitely forcible graphons. Those are graphons determined by �nitely many subgraph densities. We construct a �nitely forcible graphon such that the topological space of its typical vertices is not compact. In our construction, the space even fails to be locally compact. This disproves a conjecture of Lovasz and Szegedy.
