The k-set problem: how many halving lines can n points have?

Find the maximum number of k-sets and halving lines of n points in the plane; the known bounds are barely superlinear and O(n^4/3).

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File it in the catalog (shoal-catalog 0521c19bbd38)

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	1	+{
	2	+ "domain": "computer-science",
	3	+ "topics": [
	4	+ "computational-geometry",
	5	+ "discrete-geometry"
	6	+ ],
	7	+ "type": "proof",
	8	+ "also": [
	9	+ "construction"
	10	+ ],
	11	+ "cover": {
	12	+ "motif": "points",
	13	+ "n": 8
	14	+ },
	15	+ "source": {
	16	+ "key": "topp",
	17	+ "name": "The Open Problems Project",
	18	+ "url": "https://topp.openproblem.net/p7",
	19	+ "id": "7",
	20	+ "curator": "Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke",
	21	+ "status": "open",
	22	+ "as_of": "2026-10-20",
	23	+ "catalog": "0521c19bbd38"
	24	+ }
	25	+}