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28 March 2024
 
  » arxiv » 1002.0374

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Density Hales-Jewett and Moser numbers
D.H.J. Polymath ;
Date 2 Feb 2010
AbstractFor any $n geq 0$ and $k geq 1$, the emph{density Hales-Jewett number} $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := ${1,...,k}^n$ which contains no combinatorial line; similarly, the Moser number $c’_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson shows that $c_{n,k}$ = $o(k^n)$ as $n o infty$ (which implies a similar claim for $c’_{n,k}$); this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established.
Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c’_{n,k}$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,...,6$ is $1,2,6,18,52,150,450$, while the sequence $c’_{n,3}$ for $n=0,...,6$ is $1,2,6,16,43,124,353$. We also prove some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k = 2$ case) does not hold for higher $k$, and also establishing the asymptotic lower bound $c_{n,k} geq k^n exp(- O(sqrt[ell]{log n}))$ where $ell$ is the largest integer such that $2k > 2^ell$.
Source arXiv, 1002.0374
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