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Omni-MATH / Let S be a finite set of points in the plane. A linear partition of S is an unordered pair {A,B} of subsets of S such that A∪B=S, A∩B=∅, and A and B lie on opposite sides of some s…

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problem

Let SS be a finite set of points in the plane. A linear partition of SS is an unordered pair {A,B}\{A,B\} of subsets of SS such that A∪B=SA \cup B = S, A∩B=∅A \cap B = \emptyset, and AA and BB lie on opposite sides of some straight line disjoint from SS (AA or BB may be empty). Let LSL_S be the number of linear partitions of SS. For each positive integer nn, find the maximum of LSL_S over all sets SS of nn points.
Plain-text mathematical notation (without MathML)
Let S be a finite set of points in the plane. A linear partition of S is an unordered pair {A,B} of subsets of S such that A∪B=S, A∩B=∅, and A and B lie on opposite sides of some straight line disjoint from S (A or B may be empty). Let L_(S) be the number of linear partitions of S. For each positive integer n, find the maximum of L_(S) over all sets S of n points.
Original LaTeX notation
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points.

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Source and history

task_id
3404ec90-e992-5d01-b742-1331a5ee1c98
task_key
test--3404ec90-e992-5d01-b742-1331a5ee1c98
task_revision_id
1
config / split
/ test

Official source

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