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**11 - 15**of**15**### COMPUTABLY ENUMERABLE PARTIAL ORDERS

"... Abstract. We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the latter is strictly s ..."

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Abstract. We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the latter is strictly stronger than the latter. We then show that every ∅ ′-computable structure (or even just of c.e. degree) has the same degree spectrum as some computably enumerable (co-c.e.) partial order, and hence that there is a c.e. (co-c.e.) partial order with spectrum equal to the set of nonzero degrees. 1.

### unknown title

, 2005

"... In this paper I give a proof that NP is not equal to P and co-NP is not equal to P. I use the clique problem which is proved to be NP-complete in Karp [2]. To prove NP! = P, it is shown that clique problem does not have a polynomial time algorithm in the worst case. In §2 the various definitions, no ..."

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In this paper I give a proof that NP is not equal to P and co-NP is not equal to P. I use the clique problem which is proved to be NP-complete in Karp [2]. To prove NP! = P, it is shown that clique problem does not have a polynomial time algorithm in the worst case. In §2 the various definitions, notations, assumptions and a lemma to define definitions is

### unknown title

, 2005

"... ABSTRACT. The NP-problem is solved by showing that the clique problem has no polynomial time algorithm. It is shown that all algorithms for clique problem are of a particular type say someType. It is then proved that all algorithms of someType are not polynomial in the worst case. 1. ..."

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ABSTRACT. The NP-problem is solved by showing that the clique problem has no polynomial time algorithm. It is shown that all algorithms for clique problem are of a particular type say someType. It is then proved that all algorithms of someType are not polynomial in the worst case. 1.

### RANDOM REALS, THE RAINBOW RAMSEY THEOREM, AND ARITHMETIC CONSERVATION

, 2012

"... We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts? ” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be impleme ..."

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We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts? ” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA0 for arithmetic sentences. Thus, from the Csima-Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over RCA0 + BΣ2 for Π1 1-sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P − + I Σ1 and no stronger than P − + BΣ2.

### A ... Set With No Infinite Low Subset in . . .

, 2000

"... We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and eective content of versions of Ramsey's Theorem. ..."

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We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and eective content of versions of Ramsey's Theorem.