Title: Essential Covers of the Cube by Hyperplanes
| Speaker: | Igor Araujo |
| Affiliation: | University of Illinois Urbana-Champaign |
| Location: | contact Sabrina Lato for Zoom link |
Abstract: An essential cover of the vertices of the n-cube $\{0,1\}^n$ by hyperplanes is a minimal covering where no hyperplane is redundant, and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a construction of an essential cover with $\lceil \frac{n}{2} \rceil + 1$ hyperplanes and showed that $\Omega(\sqrt{n})$ hyperplanes are required. Recently, Yehuda and Yehudayoff improved the lower bound by showing that any essential cover of the $n$-cube contains at least $\Omega(n^{0.52})$ hyperplanes. Building on the method of Yehuda and Yehudayoff, we prove that $\Omega \left( \frac{n^{5/9}}{(\log n)^{4/9}} \right)$ hyperplanes are needed.
This is joint work with József Balogh and Letícia Mattos.