# Sh:502

- Komjáth, P., & Shelah, S. (1993).
*On uniformly antisymmetric functions*. Real Anal. Exchange,**19**(1), 218–225. arXiv: math/9308222 MR: 1268847 -
Abstract:

We show that there is always a uniformly antisymmetric f:A\to\{0,1\} if A\subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R\to\omega with |S_x|\leq 1 for every x\in R. If the continuum is at least \aleph_n then there exists a point x such that S_x has at least 2^n-1 elements. We also show that there is a function f:Q\to\{0,1,2,3\} such that S_x is always finite, but no such function with finite range on R exists - Version 1993-08-27_10 (7p)

Bib entry

@article{Sh:502, author = {Komj{\'a}th, P{\'e}ter and Shelah, Saharon}, title = {{On uniformly antisymmetric functions}}, journal = {Real Anal. Exchange}, fjournal = {Real Analysis Exchange}, volume = {19}, number = {1}, year = {1993}, pages = {218--225}, issn = {0147-1937}, mrnumber = {1268847}, mrclass = {26A15 (03E50 04A20)}, note = {\href{https://arxiv.org/abs/math/9308222}{arXiv: math/9308222}}, arxiv_number = {math/9308222} }