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  1. ; (, Advances in Combinatorics)
    For a finite point set P⊂R^d, denote by diam(P) the ratio of the largest to the smallest distances between pairs of points in P. Let c_{d,α}(n) be the largest integer c such that any n-point set P⊂R^d in general position, satisfying diam(P)<αn^{1/d}, contains an c-point convex independent subset. We determine the asymptotics of c_{d,α}(n) as n→∞ by showing the existence of positive constants β=β(d,α) and γ=γ(d) such that βn^{(d−1)/(d+1)}≤c_{d,α}(n)≤γn^{(d−1)/(d+1)} for α≥2. 
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    Free, publicly-accessible full text available January 29, 2026
  2. ; (, SIAM journal on discrete mathematics)
    We prove that every family of (not necessarily distinct) even cycles D_1,...,D_{1.2n-1} on some fixed n-vertex set has a rainbow even cycle (that is, a set of edges from distinct D_i’s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, the result is best possible for every positive integer n. 
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    Accepted Manuscript Full Text Available
  3. ; (, 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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    Full Text Available 
  4. ; (, SIAM Journal on Discrete Mathematics)
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