erdős #101

← #100 · #102 → (packet.json; erdosproblems.com)

Given $n$ points in $R^{2}$ , no five of which are on a line, the number of lines containing four points is $o (n^{2})$ .

Worked, still open.

geometry · open · prize $100 · formalized (Lean) · 0 attempts

machinery: geometric,incidence-geometry,extremal-set-system,point-line-incidences,rich-lines-upper-bound,no-k-collinear-constraint

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

This statement is **not proved** (and not disproved). It is a well-known **open problem of Erdős**. In fact it appears in lists of Erdős problems as “Given $n$ points in (\mathbb R^2), no five collinear, is the number of 4-point lines $o(n^2)$?” and is marked **OPEN**. ([Erdős Problems][1])

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 52 · open (literature)

theorem erdos_101 : (fun n => (numLinesWithFourPointMax n : ℝ)) =o[atTop] (fun n => (n : ℝ)^2)

formal-conjectures/101.lean ↗

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

links

Green's open problems list · paper

#588Let

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

?A006065 #669Let

F_{k} (n)

be minimal such that for any

n

points in

R^{2}

there exist at most

F_{k} (n)

many distinct lines passing through at least

k

of the points, and

f_{k} (n)

k

points.Estimate

f_{k} (n)

and

F_{k} (n)

- in particular, determine

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

and

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

.A006065

status

open