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SUMMARY:Hao Huang (Emory University)
DTSTART:20201008T020000Z
DTEND:20201008T030000Z
DTSTAMP:20260423T021015Z
UID:SCMSComb/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SCMSComb/18/
 ">Covering cubes by hyperplanes</a>\nby Hao Huang (Emory University) as pa
 rt of SCMS Combinatorics Seminar\n\n\nAbstract\nNote that the vertices of 
 the $n$-dimensional cube $\\{0\, 1\\}^n$ can be covered by two affine hype
 rplanes $x_1=1$ and $x_1=0$. However if we leave one vertex uncovered\, th
 en suddenly at least $n$ affine hyperplanes are needed. This was a classic
 al result of Alon and F\\"uredi\, followed from the Combinatorial Nullstel
 lensatz.\n\nIn this talk\, we consider the following natural generalizatio
 n of the Alon-F\\"uredi theorem: what is the minimum number of affine hype
 rplanes such that the vertices in $\\{0\, 1\\}^n \\setminus \\{\\vec{0}\\}
 $ are covered at least $k$ times\, and $\\vec{0}$ is uncovered? We answer 
 the problem for $k \\le 3$ and show that a minimum of $n+3$ affine hyperpl
 anes is needed for $k=3$\, using a punctured version of the Combinatorial 
 Nullstellensatz. We also develop an analogue of the Lubell-Yamamoto-Meshal
 kin inequality for subset sums\, and solve the problem asymptotically for 
 fixed $n$ and $k \\rightarrow \\infty$\, and pose a conjecture for fixed $
 k$ and large $n$.\n\nJoint work with Alexander Clifton (Emory University).
 \n\npassword 121323\n
LOCATION:https://researchseminars.org/talk/SCMSComb/18/
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