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Creators/Authors contains: "Vasileuski, Alexey"

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  1. ; (, The Electronic Journal of Combinatorics)
    Given finite sets $$X_1,\dotsc,X_m$$ in $$\mathbb{R}^d$$ (with $$d$$ fixed), we prove that there are respective subsets $$Y_1,\dotsc,Y_m$$ with $$\lvert Y_i\rvert \geq \frac{1}{poly(m)}\lvert X_i\rvert$$ such that, for $$y_1\in Y_1,\dotsc,y_m\in Y_m$$, the orientations of the\linebreak $(d+1)$-tuples from $$y_1,\dotsc,y_m$$ do not depend on the actual choices of points $$y_1,\dotsc,y_m$$. This generalizes previously known case when all the sets $$X_i$$ are equal. Furthermore, we give a construction showing that polynomial dependence on $$m$$ is unavoidable, as well as an algorithm that approximates the best-possible constants in this result. 
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