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Chapter 22. Application to discrete geometry

Chapter 22. Application to discrete geometry

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138



22. Application to discrete geometry



solved by Gallai, and popularized by Erd˝

os through his article in the

American Mathematical Monthly [31].

Theorem 22.2. Any set of m non-collinear points in the plane determine at least m lines.

The idea of the following proof goes back to de Bruijn and Erd˝os [16].

Proof. Let p1 , . . . , pm be the given m points. Suppose that they

determine n lines, and let L = {l1 , l2 , . . . , ln } be the set of the lines.

Let Ci ⊂ L be the subset of lines passing through pi , and let H =

{C1 , . . . , Cm } ⊂ 2L . Then, for all i = j, Ci ∩Cj contains just one line,

that is, the line connecting two points pi and pj . Thus |Ci ∩ Cj | = 1,

and by the Fisher’s inequality we get |H| = m ≤ |L| = n.

Chv´

atal posed a generalization of this problem in a metric space.

We say that three points x, y, and z in a metric space with metric d

are collinear1 if they satisfy

d(x, z) = d(x, y) + d(y, z).

For example, a graph can be viewed as a metric space by the usual

shortest-path metric. Then we can define a line in the graph in the

sense above. Such lines behave rather differently from the usual lines

in Euclidean space. Nevertheless, Chen and Chv´

atal conjectured the

following.

Conjecture 22.3 (Chen–Chv´

atal [18]). In any metric space, m noncollinear points determine at least m lines.



22.2. Chromatic number of the unit-distance

graph

A family of subsets H ⊂ 2[n] is called t-avoiding if

|H ∩ H | = t

for all H, H ∈ H. Applying Theorem 21.8 to the case

n = 4p − 1, s = p − 1,

we get the following.

1



This is called collinearity by betweeness.



L = {0, . . . , p − 2},



22.2. Chromatic number of the unit-distance graph

Corollary 22.4 ([67]). Let p be a prime. If a family H ⊂

(p − 1)-avoiding, then

|H| ≤



139

[4p−1]

2p−1



is



4p − 1

4p − 1

4p − 1

+

+ ··· +

.

0

1

p−1



4p−1

Exercise 22.5. Show that 4p−1

+ 4p−1

+ · · · + 4p−1

0

1

p−1 < 2 p−1 .

s

(Hint: Use induction on s to deduce that i=0 ni < 2 ns for n ≥ 3s.)



Now we present a geometric application of the above result. The

distance-d graph in n-dimensional Euclidean space has vertex set Rn

and two points are adjacent if their Euclidean distance is d. If d =

1 then it is called the unit-distance graph and denoted by Gn . In

symbols:

• V (Gn ) = Rn ,

• E(Gn ) = {{x, y} : x − y = 1},

where x = x21 + · · · + x2n for x = (x1 , . . . , xn ) ∈ Rn . Note that

distance-d graphs in a fixed dimension are isomorphic to each other

for all d > 0.

It is difficult to determine the chromatic number of Gn , and even

for the planar case (n = 2) we only know that

5 ≤ χ(G2 ) ≤ 7.

For the general case Larman and Rogers [86] proved that

χ(Gn ) < (3 + o(1))n .

On the other hand, Frankl and Wilson [67] obtained the following

lower bound using Corollary 22.4.

Theorem 22.6. Let p be a prime, and let n = 4p − 1. Then we have

χ(Gn ) > 1.1n .



Proof. Let k = 2p − 1 and d = 2p. As usual we identify 2[n] and

Ω = {0, 1}n . We are going to find a large distance-d structure in the

n-dimensional cube Ωn . To this end we look at the points having

exactly k 1’s in the coordinate, that is, we look at subsets in [n]

k . If

satisfy

two subsets X, Y ∈ [n]

k

|X ∩ Y | = p − 1,



140



22. Application to discrete geometry



then the corresponding characteristic vectors satisfy

x − y = d,

where we used x − y



2



= |X Y | = 2(k − |X ∩ Y |) = 2p.



Now suppose that we can color [n]

so that any two points with

k

distance d get different colors. Then a family of subsets having the

same color is (p − 1)-avoiding, and, by Corollary 22.4 with Exern

. So in order to

cise 22.5, the size of the family is less than 2 p−1

color the distance-d graph, or equivalently to color G2 , we need at

n

n

/(2 p−1

) > 1.1p colors2 .

least 2p−1

With a little bit more effort one can show that χ(Gn ) > 1.2n for

sufficiently large n ∈ N; see [67].



2

n



1.1 .



n

2p−1



/(2



n

p−1



)=



1 (3p)···(2p+1)

2 (2p−1)···p



=



3p (3p−1)···(2p+1)

2p (2p−1)···(p+1)



> (3/2)p = (3/2)



n+1

4



>



Chapter 23



Upper bounds using

inclusion matrices



For two hypergraphs F and G we can define an |F| × |G| matrix

M whose (F, G)-entry is determined by a given function m(F, G).

Obviously rank M ≤ min{|F|, |G|}. If the rows of M are linearly

independent, then we get rank M = |F| ≤ |G|. In this chapter we

present some applications of this inequality.



23.1. Bounds for s-independent families

[n]

For 0 ≤ i ≤ k ≤ n, F ⊂ [n]

k , and G ⊂

i , define the inclusion

matrix M (F, G) as follows. This is an |F| × |G| matrix whose (F, G)entry m(F, G), where F ∈ F and G ∈ G, is defined by



m(F, G) =

For F ⊂

(23.1)



[n]

k



1



if F ⊃ G,



0



if F ⊃ G.



and 0 ≤ j ≤ i ≤ k, simple counting yields

M (F,



[n]

i



)M (



[n]

i



,



[n]

j



)=



k−j

i−j



M (F,



[n]

j



In fact, the (F, J)-entry of (23.1), where F ∈ F and J ∈

#{I ∈



[n]

i



).

[n]

j



, counts



: J ⊂ I ⊂ F }.

141



142



23. Upper bounds using inclusion matrices



For a matrix M , the column space is a vector space spanned by its

column vectors. Let colsp M denote the column space of M . Then

(23.1) also shows the following.

Lemma 23.1. Let 0 ≤ j ≤ i ≤ k and F ⊂

colsp M (F,



[n]

j



[n]

k



) ⊂ colsp M (F,



. Then

[n]

i



).



[n]

We say that F ⊂ [n]

k is s-independent if the rows of M (F, s )

are linearly independent, that is, the inclusion matrix has full rowrank. In this case, |F| ≤ ns immediately follows.



Lemma 23.2. Let p be a prime and let f ∈ Q[x] be a polynomial of

degree s. Let F ⊂ [n]

k . Suppose that F satisfies

(23.2)



f (|F ∩ F |)



≡ 0 (mod p)



if F = F ,



≡ 0 (mod p)



if F = F



for F, F ∈ F. Then F is s-independent.

Proof. Since f is of degree s, it can be uniquely represented as

s



f (x) =



αi

i=0



x

,

i



where αi ∈ Q. Let Mi = M (F, [n]

i ) be the inclusion matrix. Then

T

the (F, F )-entry of Mi Mi counts the number of i-element subsets

|

. Let

contained in both F and F , that is, |F ∩F

i

s



(23.3)



αi Mi MiT .



A=

i=0



Then the (F, F )-entry of A is f (|F ∩F |), and A (mod p) is a diagonal

matrix with no zero diagonal entries. So A is non-singular because

det A ≡ 0 (mod p), and rank A = |F|. By Lemma 23.1 it follows that

colsp Mi ⊂ colsp Ms for 0 ≤ i ≤ s. This, together with (23.3), yields

colsp A ⊂ colsp Ms , and hence rank A ≤ rank Ms . Thus Ms has full

row-rank, that is, F is s-independent.

Theorem 23.3 (Frankl–Wilson [67]). Let p be a prime and let L ⊂

[0, p − 1] = {0, 1, . . . , p − 1} with |L| = s. Suppose that F ⊂ [n]

k



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