Smallest value of largest angle in finite planar configurations

Does every set of $n$ points in the Euclidean plane contain three points $A,B,C$ such that the two segments obtained by joining $A,B$, respectively $A,C$ form an angle at least equal to $(1-2/n)\pi$ at the point $A$? (Equality is of course achieved by the vertex set of a regular $n-$gone.)

Pietro Majer's example below can be generalized and shows that $(1-2/n)\pi$ has to be replaced by a somewhat smaller constant (at least for values of $n$ which are large enough). For his example we have to take $(1-2/6)\pi$ instead of $(1-2/7)\pi$. Is the best possible constant $(1-a(n))\pi$ asymptotically substantially better, ie. can $na(n)$ become for example arbitrarily large for $n$ large enough? (It is of course obvious that $a(n)$ is decreasing but how fast?)

Update: For $n=5$, one can get arbitrarily close to $(1-1/4)\pi$: Take a right-angled isocele triangle. Split the right-angled-vertex infinitesimally along a line parallel to the longest side of the initial triangle and add an additional point on the symmetry axis very high above the two infinitesimal points.

The key bound is $$(1 - 1/n) \pi$$, due to Erdős and Szekeres:

The above is an excerpt from this paper:

The Erdős-Szekeres result is in their 1961 paper, "On some extremum problems in elementary geometry.", Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 3-4, 53-62 (1961) (PDF download link).

• How does the configuration of $8=2^3$ points without angles greater than $2/3\pi+\epsilon$ look like? Commented Nov 14, 2012 at 12:44
• @Roland: I added a link to the Erdős-Szekeres paper. Their proof is in Section 4, p.59 ff. I cannot pursue it myself at the moment; sorry. Commented Nov 14, 2012 at 13:15
• Thank you. By the way, the configuration on $8$ points is easy: one can split the central point in Pietro Majer's example into two infinitesimally close points on a line parallel to a side of the initial triangle. Commented Nov 14, 2012 at 14:55
• actually this way we make a new angle (1−1/6)π... Commented Nov 14, 2012 at 17:39
• I am not sure whether it's just me or whether the link is dead. In any case, this link seems to work. Here is also Wayback Machine link. Commented Aug 15, 2019 at 6:58

Consider an exagon obtained from an equilateral triangle by cutting three small equilateral triangles from its vertices. Consider the configuration of the $6$ vertices of the exagon, plus the center of the initial equilateral triangle. It seems to me that with these $7$ points one can't do angles larger than $2\pi/3+\epsilon$.

• Indeed. So what is the best possible constant for a given integer $n$? Commented Nov 14, 2012 at 9:14
• Yes, it seems a beautiful and difficult variational problem. Commented Nov 14, 2012 at 10:01
• Sorry, since I can accept only one answer and since Joseph O'Rourke mention of the Erdos-Szekeres paper kills the problem completely, I have changed the accepted answer to his reply. Commented Nov 15, 2012 at 9:50