Email: abernsht ~at~ math.cmu.edu
Office: 7101 Wean Hall
Welcome!
I am a Postdoctoral Associate in the Department of Mathematical Sciences at Carnegie Mellon University.
I did my Ph.D. at the University of Illinois at UrbanaChampaign under the supervision of Alexandr Kostochka and Anush Tserunyan.
You can find my CV here (last update: June 18, 2019).
Research
My main areas of research are combinatorics and descriptive set theory. Some particular topics I am interested in include: graph coloring, probabilistic methods, extremal combinatorics, and Borel/measurable combinatorics with applications to ergodic theory.
Papers and Preprints
 With M. Delcourt and A. Tserunyan. Independent sets in algebraic hypergraphs. Preprint.
 On Baire measurable colorings of group actions. Preprint.
 With A. Kostochka and X. Zhu. Fractional DPcolorings of sparse graphs. Journal of Graph Theory (to appear).
 Measurable versions of the Lovász Local Lemma and measurable graph colorings. Advances in Mathematics (to appear).
 With M. Tait. Improved lower bound for difference bases. Journal of Number Theory (to appear).
 Ergodic theorems for the shift action and pointwise versions of the Abért–Weiss theorem. Israel Journal of Mathematics (to appear).
 Multiplication of weak equivalence classes may be discontinuous. Transactions of the American Mathematical Society (to appear).
 Building large free subshifts using the Local Lemma. Groups, Geometry, and Dynamics (to appear).
 With A. Kostochka. DPcolorings of hypergraphs. European Journal of Combinatorics (to appear).
 With O. Khormali, R.R. Martin, J. Rollin, D. Rorabaugh, S. Shan, and A.J. Uzzell. Regular colorings in regular graphs. Discussiones Mathematicae Graph Theory (to appear).
 The JohanssonMolloy Theorem for DPcoloring. Random Structures and Algorithms (2019).
 With M. Delcourt, H. Towsner, and A. Tserunyan. A short nonalgorithmic proof of the containers theorem for hypergraphs. Proceedings of the American Mathematical Society (2019).
 With A. Kostochka. On differences between DPcoloring and list coloring (in Russian). Matematicheskie Trudy (2018); English version.
 With A. Kostochka. Sharp Dirac’s theorem for DPcritical graphs. Journal of Graph Theory (2018).
 With A. Kostochka and X. Zhu. DPcolorings of graphs with high chromatic number. European Journal of Combinatorics (2017).
 The Local Cut Lemma. European Journal of Combinatorics (2017).
 With A. Kostochka and S. Pron. On DPcoloring of graphs and multigraphs (in Russian). Siberian Mathematical Journal (2017); English version.
 The asymptotic behavior of the correspondence chromatic number. Discrete Mathematics (2016).
 New bounds for the acyclic chromatic index. Discrete Mathematics (2016).
 With A. Kostochka. On the number of edges in a graph with no (k+1)connected subgraphs. Discrete Mathematics (2016).
 3Regular subgraphs and (3,1)colorings of 4regular pseudographs (in Russian). Discrete Analysis and Operations Research (2014).
 With N. Shilov. Robots in Space Multiagent Problem: complexity, information and cryptographic aspects (in Russian). Modeling and Analysis of Information Systems (2013).
Expository Writing
Teaching
Current Teaching
Research Topics in Combinatorics (21499). Course website.
Highlights of Past Teaching
At CMU:
At UIUC:
Department of Mathematical Sciences
Carnegie Mellon University
Mellon College of Science
