## Combinatorial Problems with Geometric Flavour

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## Abstract

This thesis consists of an introduction and nine chapters, each devoted to a different combinatorial problem. What gives these problems coherence is that although at first sight they look very different, the essential difficulties in most are geometric.

In Chapters 2 and 3, we investigate sumset inequalities. For sets

We prove a sharp stability result for the classical Freiman--Bilu

In particular, we deduce the sharp stability result for the Brunn--Minkowski inequality for equal sets, conjectured by Figalli and Jerison. Given

Finally, we prove a strengthen version of the aforementioned conjecture by Figalli and Jerison for a specific class of geometric objects. We find the optimal constants

In Chapters 4, 5 and 6, we investigate covering systems. A covering system is a finite collection of arithmetic progressions

Another beautiful conjecture, proposed by Erdős and Graham in 1980, states that if the moduli are distinct elements of the interval

Our method has a number of further applications. Most importantly, we prove the conjecture of Schinzel. In addition, we give an alternative (somewhat simpler) proof of a breakthrough result of Hough, who resolved Erdős' minimum modulus problem, with an improved bound on the smallest difference. Moreover, we make further progress on the problem of Erdős and Selfridge, which, in particular, we solve in the square free case.

Finally, we answer another natural question of Erdős, asked in 1952, on the \emph{number} of minimal covering systems. Addressing this question requires very different methods. %, that is, covering systems such that the removal of any progression leaves an element uncovered.
More precisely, we show that the number of minimal covering systems with exactly

Chapter 7 is devoted to Littlewood polynomials. Answering a question of Erd\H{o}s from 1957 and confirming a conjecture of Littlewood from 1966, we show that there exist absolute constants

Chapter 8 is devoted to judicious partitions. Bollobás, Reed and Thomason proved that every

Chapter 9 is devoted to coloured structures in the Boolean lattice. Given a collection of coloured chain posets, we estimate the number of coloured subsets of the Boolean lattice which avoid all chains in this collection. Our proof relies on the recent powerful method of hypergraph containers developed independently by Balogh, Morris and Samotij as well as Saxton and Thomason, inspired by the previous work of Conlon and Gowers. In order to prove results about coloured chain posets we need to apply the hypergraph container lemma recursively, with different uniformities at each stage, using a balanced supersaturation result for a certain non-uniform hypergraph encoding forbidden configurations. Our work extends to a coloured setting the previous work of Collares and Morris as well as Balogh, Mycroft, and Treglown on the number of antichains in a random subset of the Boolean lattice.

Chapter 10 is devoted to positional games. For two graphs