Abstract: Let F be some graph. Consider first the following one player game, known as balanced Ramsey game: The Player has r colors and in each round r random edges (that were never seen before) on n vertices are presented to her. Immediately, the Player has to color all these edges differently. The game ends as soon as a monochromatic copy of F appears.

Secondly consider the Achlioptas game: Here the Player only loses when she creates a copy of F in one distinguished color.

Abstract: Vu, Wood and Wood showed that any finite set S in a characteristic zero integral domain can be mapped to F_p, for infinitely many primes p, while preserving all algebraic incidences of S. In this talk we will show that the converse essentially holds, namely any small subset of F_p can be mapped to some finite algebraic extension of Q, while preserving bounded algebraic relations. This answers a question of Vu, Wood and Wood.

Most of the talk will be devoted to the presentation of applications of the main result. For small subsets of F_p, we: show that the Szemerédi-Trotter theorem holds with optimal exponent 4/3, improve the previously best-known sum-product estimate, transfer results from C to F_p concerning sets with small doubling constant, and so on.

Abstract: One of the first Maker-Breaker games that was analyzed when the concept was introduced is the Hamiltonicity game. Already Erdős and Chvátal found out that Maker can build a Hamilton cycle playing on the edge set of K_n for sufficiently large n. It took decades and several partial results of different authors before Krivelevich proved their conjecture and showed that the game in fact obeys the random graph intuition, showing that the critical bias is asymptotically log(n)/n.

In this talk, we consider the Hamiltonicity game played on the edge set of the random graph G(n, p) for some appropriate p. Confirming (and even strengthening) a conjecture of Stojaković and Szabó, we show that this game also follows the random graph intuition, that is, the critical bias is asymptotically np/log(n) a.a.s.. As far time permits, I present the main steps of the proof.

Abstract: Let F be a finite field. A subset S ⊆ Fⁿ is called (k, c)-evasive if it has intersection of size at most c with every k-dimensional affine subspace of Fⁿ. A simple probabilistic argument shows that a random set S ⊂ Fⁿ of size |F|^(1 - ε)n will have intersection of size at most O(k/ε) with any k-dimensional affine subspace. Recently, Zeev Dvir and Schachar Lovett gave an explicit construction of a (k, (k/ε)^k)-evasive set S ⊂ Fⁿ of size |F|^(1 - ε)n.

Abstract: We shall discuss a polynomial time randomized algorithm that, on receiving as input a pair (H, G) of n-vertex graphs, searches for an embedding of H into G. If H has bounded maximum degree and G is suitably dense and pseudorandom, then the algorithm succeeds with high probability. Our algorithm proves that, for every integer d ≥ 3 and suitable constant C = C_d, as n → ∞, asymptotically almost all graphs with n vertices and ⌊Cn^2 - 1/d log^1/dn⌋ edges contain as subgraphs all graphs with n vertices and maximum degree at most d.

Abstract: In an orientation game, two players called OMaker and OBreaker take turns in directing previously undirected edges of the complete graph on n vertices. The final graph is a tournament. OMaker wins if this tournament has some predefined property P, otherwise OBreaker wins. We analyse the Oriented-cycle game, in which OMaker's goal is to close a directed cycle. Since this game is drastically in favour of OMaker, we allow OBreaker to direct (up to) b > 1 edges in each of his moves and ask for the largest b* such that OMaker has a winning strategy (the so called threshold bias). It is known that n/2 - 3 < b* < n - 2, where the upper bound was conjectured to be tight.

In this talk I will discuss recent developments in the field of orientation games, including a sketch of an OBreaker strategy which improves the upper bound on b* to roughly 5n/6.

Abstract: The Blow-up Lemma is an important tool for embedding spanning graphs H into dense graphs G. One of its limitations is, however, that it can only be used if the maximum degree of H is bounded by a constant. We recently established a generalisation of this lemma which can handle even H which are only c-arrangeable (and observe a very weak maximum degree bound). Here a graph is called c-arrangeable if its vertices can be ordered in such a way that the neighbours to the right of any vertex v have at most c neighbours to the left of v in total.

In the talk I will explain this lemma, discuss several applications, and outline parts of the proof.