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erdős #588

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Let $f_{k} (n)$ be minimal such that if $n$ points in $R^{2}$ have no $k + 1$ points on a line then there must be at most $f_{k} (n)$ many lines containing at least $k$ points. Is it true that $f_{k} (n) = o (n^{2})$ for $k \geq 4$ ?

Worked, still open.

geometry · open · prize $100 · 0 attempts

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vela registry pull vfr_37aec80d874a0239

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

No one knows at present [[nomath]](even for $k=4$)[[/nomath]].

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

oeis

A006065 — Maximal number of 4-tree rows in n-tree orchard problem.0,0,0,1,1,1,2,2,3,5,6,7,9,10,12,15,16,18,20,23 A008997 — Orchard problem with 5 trees in a row (may not have all been proved optimal).0,0,0,0,1,1,1,1,2,2,2,3,3,4,6,6,7,9,10,11

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n

R^{2}

, no five of which are on a line, the number of lines containing four points is

o (n^{2})

.A006065 #669Let

F_{k} (n)

be minimal such that for any

n

R^{2}

there exist at most

F_{k} (n)

many distinct lines passing through at least

k

of the points, and

f_{k} (n)

similarly but with lines passing through exactly

k

points.Estimate

f_{k} (n)

F_{k} (n)

- in particular, determine

lim F_{k} (n) / n^{2}

lim f_{k} (n) / n^{2}

status

open

finding.noted · reviewer:will-blair · 1 day

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