Question 80
Given a positive integer $n$, a set $S$ is *n-admissible* if:
• each element of $S$ is an unordered triple of distinct integers in $\{1, 2, \dots, n\}$,
• $|S| = n-2$, and
• for each $1 \le k \le n-2$ and each choice of $k$ distinct $A_1, A_2, \dots, A_k \in S$,
$$
|A_1 \cup A_2 \cup \dots \cup A_k| \ge k + 2.
$$
Is it true that, for all $n \gt 3$ and for each $n$-admissible set $S$, there exist pairwise distinct points $P_1, \dots, P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^\circ$ for any triple $\{i, j, k\}$ in $S$?
• each element of $S$ is an unordered triple of distinct integers in $\{1, 2, \dots, n\}$,
• $|S| = n-2$, and
• for each $1 \le k \le n-2$ and each choice of $k$ distinct $A_1, A_2, \dots, A_k \in S$,
$$
|A_1 \cup A_2 \cup \dots \cup A_k| \ge k + 2.
$$
Is it true that, for all $n \gt 3$ and for each $n$-admissible set $S$, there exist pairwise distinct points $P_1, \dots, P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^\circ$ for any triple $\{i, j, k\}$ in $S$?