Gregory Igusa
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View article: On a question of Kalimullin
On a question of Kalimullin Open
We prove that for every computable limit ordinal $\alpha$ there exists a computable structure $\mathcal {A}$ which is $\Delta ^0_\alpha$-categorical and $\alpha$ is smallest such, but nonetheless for every isomorphic computable copy $\math…
View article: Some new computable structures of high rank
Some new computable structures of high rank Open
We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank , the computable infinitary theory is -categorical. Millar and Sacks asked whether this was always the case. We…
View article: The Rado Path Decomposition Theorem
The Rado Path Decomposition Theorem Open
We discuss a theorem of Rado: Every r-coloring of the pairs of natural numbers has a path decomposition.
View article: Density-1-bounding and quasiminimality in the generic degrees
Density-1-bounding and quasiminimality in the generic degrees Open
We consider the question "Is every nonzero generic degree a density-1-bounding generic degree?" By previous results \cite{I2} either resolution of this question would answer an open question concerning the structure of the generic degrees:…
View article: Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets
Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets Open
In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite $Π^0_2$ homogeneous set. The proof uses a countable collection of $Π^0_2$ sets as potential infinite homogeneous sets. …