Polynomial Extensions of the Milliken-taylor Theorem
نویسندگان
چکیده
Milliken-Taylor systems are some of the most general infinitary configurations that are known to be partition regular. These are sets of the form MT (〈ai〉i=1, 〈xn〉n=1) = { ∑m i=1 ai ∑ t∈Fi xt : F1, F2, . . . , Fm are increasing finite nonempty subsets of N}, where a1, a2, . . . , am ∈ Z with am > 0 and 〈xn〉n=1 is a sequence in N. That is, if p(y1, y2, . . . , ym) = ∑m i=1 aiyi is a given linear polynomial and a finite coloring of N is given, one gets a sequence 〈xn〉n=1 such that all sums of the form p( ∑ t∈F1 xt, . . . , ∑ t∈Fm xt) are monochromatic. In this paper we extend these systems to images of very general extended polynomials. We work with the Stone-Čech compactification βF of the discrete space F of finite subsets of N, whose points we take to be the ultrafilters on F . We utilize a simply stated result about the tensor products of ultrafilters and the algebraic structure of βF .
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