Binary subtrees with few labeled paths

Abstract

We prove several quantitative Ramseyan results involving ternary complete trees with {0,1}-labeled edges where we attempt to find a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson’s in computability theory; we show that there is a bounded Π 01 class of positive measure which is not strongly (Medvedev) reducible to DNR3; in fact, the class of 1-random reals is not strongly reducible to DNR3.

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Correspondence to Rodney G. Downey.

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Rod Downey acknowledges support from the Marsden Fund and a James Cook Fellowship. Carl Jockusch thanks the Marsden Fund for partial financial support for his travel to Wellington when the research for this paper started. Noam Greenberg also acknowledges support from the Marsden Fund. Kevin Milans acknowledges support of the National Science Foundation through a fellowship funded by the grant “EMSW21-MCTP: Research Experience for Graduate Students” (NSF DMS 08-38434).

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Downey, R.G., Greenberg, N., Jockusch, C.G. et al. Binary subtrees with few labeled paths. Combinatorica 31, 285 (2011). https://doi.org/10.1007/s00493-011-2634-3

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Mathematics Subject Classification (2000)

  • 05D99
  • 03D30