scholarly journals RAMSEY GROWTH IN SOME NIP STRUCTURES

2019 ◽  
pp. 1-29
Author(s):  
Artem Chernikov ◽  
Sergei Starchenko ◽  
Margaret E. M. Thomas
Keyword(s):  
Upper Bound ◽  
American Mathematical Society ◽  
Mathematical Journal ◽  
Mathematical Society ◽  
Duke Mathematical Journal ◽  
Combinatorial Characterization ◽  
Polynomial Boundedness ◽  
Ramsey’S Theorem ◽  
Definable Relations

We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal 163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$ -minimal expansions of $\mathbb{R}$ , and show that it does not hold in $\mathbb{R}_{\exp }$ . This provides a new combinatorial characterization of polynomial boundedness for $o$ -minimal structures. We also prove an analog for relations definable in $P$ -minimal structures, in particular for the field of the $p$ -adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society 366(9) (2014), 5043–5065), we show that in distal structures the upper bound for $k$ -ary definable relations is given by the exponential tower of height $k-1$ .

scholarly journals Places of Near-Fields: To Heinrich Wefelscheid

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Christian Karpfinger
Keyword(s):  
Quadratic Forms ◽  
Near Field ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Fundamental Result ◽  
Field Case

AbstractWefelscheid (Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg, 1971) generalised the well-known Theorem of Artin/Schreier about the characterization of formally real fields and the fundamental result of Baer/Krull to near-fields. In the last fifty years arose from the Theorem of Baer/Krull a theory, which analyses the entirety of the orderings of a field (E. Becker, L. Bröcker, M. Marshall et al.), as presented e.g. in the book by Lam (Orderings, valuations and quadratic forms, American Mathematical Society, Providence, 1983). At the centre of this theory are preorders and their compatibility with valuations or places. We develop some essential results of this theory for the near-field case. In particular, we derive the Brown/Marshall’s inequalities and Bröcker’s Theorem on the trivialisation of fans in the near-field case.

scholarly journals Equilateral stick number of knots

2014 ◽  
Vol 23 (07) ◽  
pp. 1460008 ◽  
Author(s):  
Hyoungjun Kim ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
American Mathematical Society ◽  
Upper Bounds ◽  
Crossing Number ◽  
Equal Length ◽  
Mathematical Society ◽  
Geometric Objects

An equilateral stick number s=(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55–76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s=(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s=(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s=(K1♯K2) ≤ 2c(K1) + 2c(K2).

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