Some low distortion metric ramsey problems

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

نتاج البحث: نشر في مجلةمقالةمراجعة النظراء


In this note we consider the metric Ramsey problem for the normed spaces ℓp. Namely, given some 1 ≤ p ≤ ∞ and α ≥ 1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace which embeds into ℓp with distortion at most α. In [1] it is shown that in the case of ℓ2, the dependence of m on α undergoes a phase transition at α = 2. Here we consider this problem for other ℓp, and specifically the occurrence of a phase transition for p ≠ 2. It is shown that a phase transition does occur at α = 2 for every p ∈ [1, 2]. For p > 2 we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every 1 < p < ∞ there are arbitrarily large metric spaces, no four points of which embed isometrically in ℓp.

اللغة الأصليةالإنجليزيّة
الصفحات (من إلى)27-41
عدد الصفحات15
دوريةDiscrete and Computational Geometry
مستوى الصوت33
رقم الإصدار1
المعرِّفات الرقمية للأشياء
حالة النشرنُشِر - يناير 2005
منشور خارجيًانعم


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