University of Birmingham > Talks@bham > Combinatorics and Probability seminar > Covering real grids with multiplicity

Covering real grids with multiplicity

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If you have a question about this talk, please contact Dr Richard Mycroft.

Given a field $\mathbb{F}$, finite subsets $S_1,\dots,S_d\subseteq \mathbb{F}$, and a point $\vec{p}\in S_1\times \dots\times S_d$, what is the smallest number of hyperplanes needed to cover all points of $S_1\times\dots\times S_d\setminus\{\vec{p}\}\subseteq \mathbb{F}d$ while omitting $\vec{p}$? This question was answered precisely in a celebrated paper of Alon and F\”uredi.

We will discuss a variant of this problem in which every point in $S_1\times\dots\times S_d\setminus\{\vec{p}\}$ must be covered \emph{at least $k\geq 1$ times}, while the remaining point $\vec{p}$ is again left uncovered. In contrast to the case $k=1$, this problem is generally not well understood for larger $k$. Recently Clifton and Huang and Sauermann and Wigderson investigated the special case where $\mathbb{F} = \mathbb{R}$ and the grid is $\{0,1\}d$. A natural next step is to consider larger grids over the reals. In this talk, we will focus on the two-dimensional setting and determine the answer for several different types of grids.

This is joint work with Anurag Bishnoi, Shagnik Das, and Yvonne den Bakker.

This talk is part of the Combinatorics and Probability seminar series.

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