The right acute angles problem?

The Danzer–Grünbaum acute angles problem asks for the largest size of a set of points in Rd that determines only acute angles. There has been a lot of progress recently due to the results of the second author and of Gerencsér and Harangi, and now the problem is essentially solved. In this note, we suggest the following variant of the problem, which is one way to “save” the problem. Let F(α)=limd→∞f(d,α)1∕d, where f(d,α) is the largest number of points in Rd with no angle greater than or equal to α. Then the question is to find c≔limα→π∕2−F(α). It is an intriguing question whether c is equal to 2 as one may expect in view of the result of Gerencsér and Harangi. In this paper we prove the lower bound c⩾2. We also solve a related problem of Erdős and Füredi on the “stability” of the acute angles problem and refute another conjecture stated in the same paper.