اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPoint sets with no four collinear and no large visible island
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Given a finite point set $P$ in the plane, a subset $S subseteq P$ is called an island in $P$ if $conv(S) cap P = S$. We say that $Ssubset P$ is a visible island if the points in $S$ are pairwise visible and $S$ is an island in $P$. The famous Big-line Big-clique Conjecture states that for any $k geq 3$ and $ell geq 4$, there is an integer $n = n(k,ell)$, such that every finite set of at least $n$ points in the plane contains $ell$ collinear points or $k$ pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by constructing arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size $2^{42}$.
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