Geodesic
Ideal version of Ramsey's theorem
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 289-308
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We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems. J. Combin. Theory Ser. A 54 (1990), 95–111].
We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems. J. Combin. Theory Ser. A 54 (1990), 95–111].
DOI : 10.1007/s10587-011-0073-3
Classification : 05A17, 05D10, 11B05, 40A35, 54A20
Keywords: ideal of subsets of natural numbers; Bolzano-Weierstrass theorem; Bolzano-Weierstrass property; ideal convergence; statistical density; statistical convergence; subsequence; monotone sequence; Ramsey's theorem 
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Filipów, Rafał; Mrożek, Nikodem; Recław, Ireneusz; Szuca, Piotr. Ideal version of Ramsey's theorem. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 289-308. doi: 10.1007/s10587-011-0073-3

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