De Bruijn–Erdős theorem

De Bruijn–Erdős theorem

\PMlinkescapephrase

finite plane

De Bruijn–Erdős theorem

Let

S

be a set of

n

≠

0

‘points’, say

{

L

1

,

L

2

,

…

⁢

L

n

}

, and
let

Λ

1

,

Λ

2

, …

Λ

ν

be

ν

different subsets
of

S

of which any two have exactly one point in common. Now the

De Bruijn–Erdős theorem

says that

ν

⩽

n

, and that if

ν

=

n

then (up to renumbering) at least one of the following three must be true:

•

Λ

i

=

{

L

i

,

L

n

}

for

i

≠

n

, and

Λ

n

=

{

L

n

}

•

Λ

i

=

{

L

i

,

L

n

}

for

i

≠

n

, and

Λ

n

=

{

L

1

,

L

2

,

…

⁢

L

n

-

1

}

•

n

is of the form

q

2

+

q

+

1

, each

Λ

i

contains

q

+

1

points,
and each point is contained in

q

+

1

of the

Λ

i

and we recognise the last case as

http://planetmath.org/node/6943

finite projective planes

of order

q

. For

n

=

1

(

q

=

0

) the first and last case overlap, and for

n

=

3

(

q

=

1

) the last two cases do. The second case is also known as a

near-pencil

. The last two cases together are examples of

linear spaces

.

To exclude the first two cases one usually defines

projective planes

to satisfy some non-triviality conditions; unfortunately that also excludes projective planes of order 0 and 1.

The formulation above was found in the literature

[

Cam94

]

. Naturally, if
the

L

i

are points we tend to interpret the subsets

Λ

ι

as lines. However, interpreted in this way the condition that every two

Λ

share an

L

says rather more than is needed for the structure to
be a finite plane (

http://planetmath.org/node/3509

linear space) as it insists that no two lines are

parallel

. The absence of the dual condition, for two

L

to share a

Λ

, actually means we have

too little

for the structure to be a finite plane (linear space), as for two points we may not have a line through them. And indeed, while the second and third cases are finite planes without

parallel lines

, the first case is not a plane.

Erdős–De Bruijn theorem

A dual formulation (which we could whimsically call the

Erdős–

De
Bruijn Theorem

) remedies both under- and over-specification for a plane.
Indeed, the conditions are now exactly tailored to make the structure a finite
plane (with some parallel lines in the first of the three cases).

Let

Σ

be a set of

ν

≠

0

‘points’, say

{

Λ

1

,

Λ

2

,

…

⁢

Λ

ν

}

, and let

L

1

,

L

2

, …

L

n

be

n

subsets of

Σ

(‘lines’) such that for any two points there is a unique

L

i

that contains
them both. We must now be careful about the points (the former lines) being
“different” (this was easier to formulate in the previous version, in which
form it was a

simple

incidence structure

): this condition now takes the
form that no two points must have the same collection of lines that are

incident

with them. Then

n

⩾

ν

, and if

n

=

ν

then (up to renumbering)
at least one of the following three must be true:

•

L

i

=

{

Λ

i

}

for

i

≠

n

, and

L

n

=

{

Λ

1

,

Λ

2

,

…

⁢

Λ

n

}

•

L

i

=

{

Λ

i

,

Λ

n

}

for

i

≠

n

, and

L

n

=

{

Λ

1

,

Λ

2

⁢

Λ

n

-

1

}

•

n

is of the form

q

2

+

q

+

1

, each

L

i

contains

q

+

1

points,
and each point is contained in

q

+

1

of the

L

i

For a proof of the theorem (in the former version) see e.g. Cameron

[

Cam94

]

.

References

1

Cam94

Peter J. Cameron

,

Combinatorics

: topics, techniques, algorithms

,

Camb. Univ. Pr. 1994,

ISBN  0 521 45761 0

http://www.maths.qmul.ac.uk/ pjc/comb/http://www.maths.qmul.ac.uk/ pjc/comb/

(

solutions

, errata &c.)

Title

De Bruijn–Erdős theorem

Canonical name

DeBruijnErdHosTheorem

Date of creation

2013-03-22 15:11:21

Last modified on

2013-03-22 15:11:21

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

7

Author

marijke (8873)

Entry type

Theorem

Classification

msc 05B25

Related topic

FiniteProjectivePlane4

Related topic

IncidenceStructures

Related topic

Geometry

Related topic

LinearSpace2

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