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- W3098671307 abstract "We prove a uniform version of the finite density Halpern-L{a}uchli Theorem. Specifically, we say that a tree $T$ is homogeneous if it is uniquely rooted and there is an integer $bgeq 2$, called the branching number of $T$, such that every $tin T$ has exactly $b$ immediate successors. We show the following. For every integer $dgeq 1$, every $b_1,...,b_dinmathbb{N}$ with $b_igeq 2$ for all $iin{1,...,d}$, every integer $kmeg 1$ and every real $0<epsilonleq 1$ there exists an integer $N$ with the following property. If $(T_1,...,T_d)$ are homogeneous trees such that the branching number of $T_i$ is $b_i$ for all $iin{1,...,d}$, $L$ is a finite subset of $mathbb{N}$ of cardinality at least $N$ and $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying [|Dcap big(T_1(n)times ...times T_d(n)big)| geq epsilon |T_1(n)times ...times T_d(n)|] for every $nin L$, then there exist strong subtrees $(S_1,...,S_d)$ of $(T_1,...,T_d)$ of height $k$ and with common level set such that the level product of $(S_1,...,S_d)$ is contained in $D$. The least integer $N$ with this property will be denoted by $UDHL(b_1,...,b_d|k,epsilon)$. The main point is that the result is independent of the position of the finite set $L$. The proof is based on a density increment strategy and gives explicit upper bounds for the numbers $UDHL(b_1,...,b_d|k,epsilon)$." @default.
- W3098671307 created "2020-11-23" @default.
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- W3098671307 date "2012-02-17" @default.
- W3098671307 modified "2023-09-26" @default.
- W3098671307 title "Dense Subsets of Products of Finite Trees" @default.
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- W3098671307 doi "https://doi.org/10.1093/imrn/rns015" @default.
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