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Improved bounds for the Brown-Erdos-Sos problem Conlon, David
Description
Let $f_r(n, v, e)$ be the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which contains no induced subgraph with $v$ vertices and at least $e$ edges. The Brown--Erd\H{o}s--S\'os problem of determining $f_r(n, v, e)$ is a central question in extremal combinatorics, with surprising connections to a number of seemingly unrelated areas. For example, the result of Ruzsa and Szemer\'edi that $f_3(n, 6, 3) = o(n^2)$ implies Roth's theorem on the existence of $3$-term arithmetic progressions in dense subsets of the integers. As a generalisation of this result, it is conjectured that $$f_r(n, e(r-k) + k + 1, e) = o(n^k)$$ for any fixed $r > k \geq 2$ and $e \geq 3$. The best progress towards this conjecture, due to S\'ark\"ozy and Selkow, says that $$f_r(n, e(r-k) + k + \lfloor \log e\rfloor, e) = o(n^k),$$ where the $\log$ is taken base two. In this talk, we will discuss a recent improvement to this bound.
Item Metadata
Title |
Improved bounds for the Brown-Erdos-Sos problem
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-09-04T13:43
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Description |
Let $f_r(n, v, e)$ be the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which contains no induced subgraph with $v$ vertices and at least $e$ edges. The Brown--Erd\H{o}s--S\'os problem of determining $f_r(n, v, e)$ is a central question in extremal combinatorics, with surprising connections to a number of seemingly unrelated areas. For example, the result of Ruzsa and Szemer\'edi that $f_3(n, 6, 3) = o(n^2)$ implies Roth's theorem on the existence of $3$-term arithmetic progressions in dense subsets of the integers. As a generalisation of this result, it is conjectured that
$$f_r(n, e(r-k) + k + 1, e) = o(n^k)$$
for any fixed $r > k \geq 2$ and $e \geq 3$. The best progress towards this conjecture, due to S\'ark\"ozy and Selkow, says that
$$f_r(n, e(r-k) + k + \lfloor \log e\rfloor, e) = o(n^k),$$
where the $\log$ is taken base two. In this talk, we will discuss a recent improvement to this bound.
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Extent |
32.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: California Institute of Technology
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Series | |
Date Available |
2020-03-03
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0388841
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International