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• W2772468309 abstract "In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer $nge 3$, we define $e(n)$ to be the minimum positive integer such that any set of $e(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, ErdH{o}s and Szekeres proved that $e(n) le {2n-4 choose n-2}+1$ and later in 1961, they obtained the lower bound $2^{n-2}+1 le e(n)$, which they conjectured to be optimal. We prove that $e(n) le {2n-5 choose n-2}-{2n-8 choose n-3}+2$. In a recent breakthrough, Suk proved that $e(n) le 2^{n+Oleft(n^{2/3}log nright)}$. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let $c(n)$ (and $c'(n)$) denote the smallest positive integer $N$ such that any family of $N$ pairwise disjoint convex bodies in general position (resp., $N$ convex bodies in general position, any pair of which share at most two boundary points) has an $n$ members in convex position. We show that $c(n)le c'(n)le 2^{n+Oleft(sqrt{nlog n}right)}$. Given a point set $P$ in the plane, an ordinary circle for $P$ is defined as a circle containing exactly three points of $P$. We prove that any set of $n$ points in the plane, not all on a line or a circle, determines at least $frac{1}{4}n^2-O(n)$ ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large $n$, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve $Csubset mathbb{C}^2$ of degree $d$ and two finite sets $A,Bsubset mathbb{C}$, we have $|Ccap (Atimes B)|=O_d(|A|+|B|)$. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection $|Xcap (Ptimes Q)|$ for a variety $Xsubset mathbb{C}^4$ and finite sets $P,Qsubset mathbb{C}^2$. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that $max { |A+A|, |f(A, A)|}gg |A|^{6/5}$ for any small set $Asubset mathbb{F}_p$ and quadratic non-degenerate polynomial $f(x, y)in mathbb{F}_p[x, y]$. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term $max { |A+A|, |A cdot A |}$. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on $max{ |A+B|, |f(B,C)|}$, for a specific type of function $f$. Finally, we prove that the number of distinct cubic distances generated by any small set $Atimes Asubset mathbb{F}_p^2$ is $Omega(|A|^{8/7})$, which improves a result of Yazici et al." @default.
• W2772468309 created "2017-12-22" @default.
• W2772468309 creator A5001123077 @default.
• W2772468309 date "2017-01-01" @default.
• W2772468309 modified "2023-09-23" @default.
• W2772468309 title "On some algebraic and extremal problems in discrete geometry" @default.
• W2772468309 doi "https://doi.org/10.5075/epfl-thesis-8115" @default.
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