<p>In 1948, De Bruijn and Erdős [1] proved that a finite linear space on <span class="math-container">$n$</span> points has at least <span class="math-container">$n$</span> lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane [2].</p>
<p><strong>Does a similar relation hold between <a href="https://en.wikipedia.org/wiki/Partial_linear_space" rel="nofollow noreferrer">partial linear spaces</a> and <a href="https://ufdcimages.uflib.ufl.edu/UF/00/09/93/58/00001/partialprojectiv00dows.pdf" rel="nofollow noreferrer">partial projective planes</a>?</strong></p>
<p>I guess this is a long shot; I would be happy to see any partial results.</p>
<p>PS: I'm not sure if the tag is the right one.</p>
<p>[1] <em>de Bruijn, N. G.; Erdős, Pál</em>, On a combinatorial problem, Proc. Akad. Wet. Amsterdam 51, 1277-1279 (1948). <a href="https://zbmath.org/?q=an:0032.24405" rel="nofollow noreferrer">ZBL0032.24405</a>.<br />
[2] <em>Erdős, Paul; Mullin, R. C.; Sós, Vera T.; Stinson, D. R.</em>, <a href="http://dx.doi.org/10.1016/0012-365X(83)90071-7" rel="nofollow noreferrer"><strong>Finite linear spaces and projective planes</strong></a>, Discrete Math. 47, 49-62 (1983). <a href="https://zbmath.org/?q=an:0521.51005" rel="nofollow noreferrer">ZBL0521.51005</a>.</p>

CC BY-SA 4.0
2022-08-13T16:58:10.347Z
It doesn't seem like we can compare the number of lines in a partial linear space to $n$, since it could be either a full linear space with $>n$ lines, or we could delete lines arbitrarily from a finite projective plane to get a partial linear space with $<n$ lines.
GMB
CC BY-SA 4.0
2022-08-14T05:21:19.697Z
@GMB Okay. But, I am okay with forcing more restrictions on the partial linear space. Suppose we demamnd that (i) it should be order (n,n) for some n, and (ii) #lines = #points. Now, can we say that the partial linear space must be a partial projective plane except for a finite number of cases?
Cyriac Antony
<cg.comp-geom>
Partial linear spaces and partial projective planes
Cyriac Antony
# Partial linear spaces and partial projective planes

(Asked by: username_0 on 2022-08-09)

In 1948, De Bruijn and Erdős [1] proved that a finite linear space on $n$ points has at least $n$ lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane [2].

**Does a similar relation hold between [partial linear spaces](https://en.wikipedia.org/wiki/Partial_linear_space) and [partial projective planes](https://ufdcimages.uflib.ufl.edu/UF/00/09/93/58/00001/partialprojectiv00dows.pdf)?**

I guess this is a long shot; I would be happy to see any partial results.

PS: I'm not sure if the tag is the right one.

[1] *de Bruijn, N. G.; Erdős, Pál*, On a combinatorial problem, Proc. Akad. Wet. Amsterdam 51, 1277-1279 (1948). [ZBL0032.24405](https://zbmath.org/?q=an:0032.24405).  
[2] *Erdős, Paul; Mullin, R. C.; Sós, Vera T.; Stinson, D. R.*, [**Finite linear spaces and projective planes**](http://dx.doi.org/10.1016/0012-365X(83)90071-7), Discrete Math. 47, 49-62 (1983). [ZBL0521.51005](https://zbmath.org/?q=an:0521.51005).

**Tags:** cg.comp-geom
> **Comments:**
> **username_1**: It doesn't seem like we can compare the number of lines in a partial linear space to $n$, since it could be either a full linear space with $>n$ lines, or we could delete lines arbitrarily from a finite projective plane to get a partial linear space with $<n$ lines.  
> **username_0**: @GMB Okay. But, I am okay with forcing more restrictions on the partial linear space. Suppose we demamnd that (i) it should be order (n,n) for some n, and (ii) #lines = #points. Now, can we say that the partial linear space must be a partial projective plane except for a finite number of cases?  

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## 0 answers

