The knaster problem: More counterexamples

Given a continuous function\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f:\mathbb{S}^{n - 1} \to \mathbb{R}^m $$ \end{document} andn −m + 1 pointsp1, …,pn−m + 1 ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$p_1 ,...,p_{n - m + 1} \in \mathbb{S}^{n - 1} $$ \end{document}, does there exist a rotation ϱ εSO(n) such thatf(ϱ(p1)) = … =f(ϱ(pn−m+1))? We give a negative answer to this question form = 1 ifn ε {61, 63, 65} orn≥67 and form=2 ifn≥5.