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2. Mixing properties of colourings of the ℤ ^d lattice

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   Alon, Noga; Briceño, Raimundo; Chandgotia, Nishant; Magazinov, Alexander; Spinka, Yinon (May 2021, Combinatorics, Probability and Computing)

   Abstract We study and classify proper q -colourings of the ℤ d lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $$q\le d+1$$ , there exist frozen colourings, that is, proper q -colourings of ℤ d which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $$q\ge d+2$$ , any proper q -colouring of the boundary of a box of side length $$n \ge d+2$$ can be extended to a proper q -colouring of the entire box. (3) When $$q\geq 2d+1$$ , the latter holds for any $$n \ge 1$$ . Consequently, we classify the space of proper q -colourings of the ℤ d lattice by their mixing properties. 

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3. Product of Simplices and sets of positive upper density in ℝ ^d

   https://doi.org/10.1017/S0305004117000184  

   LYALL, NEIL; MAGYAR, ÁKOS (July 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We establish that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δ k 1 and Δ k 2 are two fixed non-degenerate simplices of k 1 + 1 and k 2 + 1 points respectively, then any subset of ℝ d of positive upper Banach density with d ⩾ k 1 + k 2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δ k 1 × Δ k 2 . A new direct proof of the fact that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented. 

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