Publications.

Double jumps of linear orders

With Julia Knight and David Gonzalez

Joins of $𝒦$-pairs

With Mariya Soskova

Unions of spectra and local structure spectra

Matthew Harrison-Trainor and Karthik Ravishankar

The power of enumerations

Karthik Ravishankar

Contrasting the Halves of an Ahmad Pair

Karthik Ravishankar

We study Ahmad pairs in the ${\mathrm{\Sigma }}_2^0$ enumeration degrees. The left halves are precisely the degrees that are ${\text{low}}_3$ and join irreducible. We prove that every right half is ${\text{high}}_2$. We also characterize the left halves of Ahmad $n$-pairs using a hierarchy of join irreducibility notions. These results have consequences for the $\forall \exists$ theory of the local structure of the enumeration degrees.

List Distinguishing Number of $p^{\text{th}}$ Power of Hypercube and Cartesian Powers of a Graph

L. Sunil Chandran, Sajith Padinhatteeri, Karthik Ravi Shankar

The list distinguishing number $D_l\left(G\right)$ measures how many colors each vertex’s allowed list must contain to guarantee a coloring that breaks every nontrivial symmetry. We show that list restrictions do not increase the distinguishing number of Cartesian powers $G^r$ for connected Cartesian-prime graphs and $r\ge 3$, and determine $D_l\left(Q_n^{\left[p\right]}\right)$ for powers of hypercubes with $n\ge 7$ and $p\ge 1$.