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Rainbow arithmetic progressions and extremal subsets of lattices
Voborník, Jan ; Šámal, Robert (advisor) ; Pangrác, Ondřej (referee)
When numbers $1,\ldots,tn$ are colored with $t$ colors (each color is used $n$ times), there exists a rainbow arithmetic progression of length $k$ (rainbow progression is a progression whose terms are colored with pairwise distinct colors). This holds true for $t>k^3$. Let $T_k$ denote the smallest $t$ for which it applies. Jungic et al. conjectured $T_k=O(k^2)$. Problem relates to extremal problems in discrete hypercubes. We present a method which uses lattices (discrete hypercubes which can contain indistinguishable elements) which can lead to improving the upper bound of $T_k$ down to $O(k^2\log k)$. In this thesis, we solve several extremal problems in lattices which have corollaries in various branches of mathematics. For example, using lattices we solve edge isoperimetric inequality in Hammilton cube, we find a graph with maximal sum of squares of degrees and convex set $M\subseteq [0,b]\times[0,a]$ which maximizes function $G(M)=\int_{x=0}^a \lambda_1(M_x)^2+\int_{y=0}^b \lambda_1(M_y)^2$. Powered by TCPDF (www.tcpdf.org)
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Roth's theorem on arithmetic progressions
Krkavec, Michal ; Klazar, Martin (advisor) ; Kráľ, Daniel (referee)
Title: Roth's theorem on arithmetic progressions Author: Michal Krkavec Department: Department of Applied Mathematics Supervisor: doc. RNDr. Martin Klazar, Dr., Department of Applied Mathematics Abstract: In the presented summary work we study sets of natural numbers not containing arithmetic progressions. The aim of this thesis is to provide an overview and comparison of both analytical and combinatorial proofs of Roth's theorem, which states that every set of positive upper asymptotic density contains arithme- tic progression of length three. We also focus on the Erd˝os-Turán conjecture and Szemerédi's theorem, which finally settled the conjecture for arithmetic progres- sions of arbitrary length k. In the end, we introduce the bounds for the number r3(n), which corresponds to the largest size of a subset A ⊆ [n], which contains no arithmetic progressions of length three. At the end we present two constructions of progression-free sets. Keywords: Additive number theory, Arithmetic progressions, Roth's theorem, Elkin's construction 1
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Rainbow arithmetic progressions and extremal subsets of lattices
Voborník, Jan ; Šámal, Robert (advisor) ; Pangrác, Ondřej (referee)
When numbers $1,\ldots,tn$ are colored with $t$ colors (each color is used $n$ times), there exists a rainbow arithmetic progression of length $k$ (rainbow progression is a progression whose terms are colored with pairwise distinct colors). This holds true for $t>k^3$. Let $T_k$ denote the smallest $t$ for which it applies. Jungic et al. conjectured $T_k=O(k^2)$. Problem relates to extremal problems in discrete hypercubes. We present a method which uses lattices (discrete hypercubes which can contain indistinguishable elements) which can lead to improving the upper bound of $T_k$ down to $O(k^2\log k)$. In this thesis, we solve several extremal problems in lattices which have corollaries in various branches of mathematics. For example, using lattices we solve edge isoperimetric inequality in Hammilton cube, we find a graph with maximal sum of squares of degrees and convex set $M\subseteq [0,b]\times[0,a]$ which maximizes function $G(M)=\int_{x=0}^a \lambda_1(M_x)^2+\int_{y=0}^b \lambda_1(M_y)^2$. Powered by TCPDF (www.tcpdf.org)
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Roth's theorem on arithmetic progressions
Krkavec, Michal ; Klazar, Martin (advisor) ; Kráľ, Daniel (referee)
Title: Roth's theorem on arithmetic progressions Author: Michal Krkavec Department: Department of Applied Mathematics Supervisor: doc. RNDr. Martin Klazar, Dr., Department of Applied Mathematics Abstract: In the presented summary work we study sets of natural numbers not containing arithmetic progressions. The aim of this thesis is to provide an overview and comparison of both analytical and combinatorial proofs of Roth's theorem, which states that every set of positive upper asymptotic density contains arithme- tic progression of length three. We also focus on the Erd˝os-Turán conjecture and Szemerédi's theorem, which finally settled the conjecture for arithmetic progres- sions of arbitrary length k. In the end, we introduce the bounds for the number r3(n), which corresponds to the largest size of a subset A ⊆ [n], which contains no arithmetic progressions of length three. At the end we present two constructions of progression-free sets. Keywords: Additive number theory, Arithmetic progressions, Roth's theorem, Elkin's construction 1
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