Bounds on multiplicities and polynomial boundedness

1974 ◽  
Vol 9 (6) ◽  
pp. 1802-1809 ◽  
Author(s):  
N. N. Khuri
Keyword(s):  
Polynomial Boundedness

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 $k-1$ .

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