Open, the site's label (OPEN). Four claim pages are recorded, each answering one value of $n$ yes: $n=5$ on Pomerance's claim page (1983), a construction Erdős describes in [Er83c], which stays claimed because that transcript is its only publication; $n=7$ on Palásti's 1987 claim page (1987); $n=6$, with the further restrictions that no point is equidistant from three others and no triangle is equilateral, on Palásti's Discrete Mathematics claim page (1989); and $n=8$ on Palásti's lattice-point claim page (1989), the last three accepted on their refereed publications. With the four-point example, an isosceles triangle and its circumcenter ([Er83c], p. 53; Erdős's own example, recorded in prose rather than on a claim page), the answer is yes for $4\le n\le8$ and open for every $n\ge9$. The property does not pass to subsets, so each construction settles only its own $n$. Erdős believed that no example exists for all large $n$; this would follow from $h(n)\ge n$ for large $n$, with $h$ the function of Problem 98.