TITLE: Smallest value of largest angle in finite planar configurations
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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).