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- W2101281054 abstract "According to a classical result of Szemeredi, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to Furstenberg and Katznelson says that every dense subset of ${1,2,...,N}^d$ contains an arbitrary large grid, if $N$ is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval $[0,L]$ on the line contains an arbitrary long approximate arithmetic progression, if $L$ is large enough. (ii) every dense separated set of points in the $d$-dimensional cube $[0,L]^d$ in $RR^d$ contains an arbitrary large approximate grid, if $L$ is large enough. A further generalization for any finite pattern in $RR^d$ is also established. The separation condition is shown to be necessary for such results to hold. In the end we show that every sufficiently large point set in $RR^d$ contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case." @default.
- W2101281054 created "2016-06-24" @default.
- W2101281054 creator A5091313409 @default.
- W2101281054 date "2010-04-09" @default.
- W2101281054 modified "2023-09-27" @default.
- W2101281054 title "Approximate Euclidean Ramsey theorems" @default.
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