Deep Cliques in Point Sets

Let \(n \in \mathbb {N}\) and \(k \in \mathbb {N}_0\). Given a set P of n points in the plane, a pair \(\{p,q\}\) of points in P is called k-deep, if there are at least k points from P strictly on each side of the line spanned by p and q. A k-deep clique is a subset of P with all its pairs k-deep. We show that if P is in general position (i.e., no three points on a line), there is a k-deep clique of size at least \( \max \{1,\lfloor \frac{n}{k+1} \rfloor \}\); this is tight, for example in convex position. A k-deep clique in any set P of n points cannot have size exceeding \(n-\lceil \frac{3k}{2} \rceil \); this is tight for \(k \le \frac{n}{3}\). Moreover, for \(k \le \lfloor \frac{n}{2} \rfloor - 1\), a k-deep clique cannot have size exceeding \(2\sqrt{n(\lfloor \frac{n}{2} \rfloor -k)}\); this is tight within a constant factor. We also pay special attention to \((\frac{n}{2}-1)\)-deep cliques (for n even), which are called halving cliques. These have been considered in the literature by Khovanova and Yang, 2012, and they play a role in the latter bound above. Every set P in general position with a halving clique Q of size m must have at least \(\lfloor \frac{(m-1)(m+3)}{2}\rfloor \) points. If Q is in convex position, the set P must have size at least \(m(m-1)\). This is tight, i.e., there are sets \(Q_m\) of m points in convex position which can be extended to a set of \(m(m-1)\) points where \(Q_m\) is a halving clique. Interestingly, this is not the case for all sets Q in convex position (even if parallel connecting lines among point pairs in Q are excluded).