On the balanced upper chromatic number of cyclic projective planes and projective spaces

Discrete Mathematics 338 (2015) 2562–2571

Contents lists available at ScienceDirect

Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

On the balanced upper chromatic number of cyclic projective planes and projective spaces Gabriela Araujo-Pardo a , György Kiss b , Amanda Montejano c a

Instituto de Matemáticas, Universidad Nacional Autonóma de México, Campus Juriquilla, Querétaro, Mexico

b

Department of Geometry and MTA-ELTE GAC Research Group, Eötvös Loránd University, 1117 Budapest, Pázmány s. 1/c, Hungary

c

UMDI-Facultad de Ciencias, Universidad Nacional Autonóma de México, Campus Juriquilla, Querétaro, Mexico

article

info

Article history: Received 18 December 2014 Received in revised form 14 June 2015 Accepted 18 June 2015 Available online 17 July 2015 Keywords: Balanced rainbow-free colorings Upper chromatic number Projective planes Projective spaces

abstract We study vertex colorings of hypergraphs, such that all color sizes differ at most in one (balanced colorings) and each edge contains at least two vertices of the same color (rainbowfree colorings). Given a hypergraph H, the maximum k, such that there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number denoted by χ b (H ). Concerning hypergraphs defined by projective spaces, bounds on the balanced upper chromatic number and constructions of rainbow-free colorings are given. For cyclic projective planes of order q we prove that:

q2 + q + 1 6

≤ χ b (Πq ) ≤

q2 + q + 1 3

.

We also give bounds for the balanced upper chromatic numbers of the hypergraphs arising from the n-dimensional finite space PG(n, q). © 2015 Elsevier B.V. All rights reserved.

1. Introduction The notion of a rainbow-free coloring [11] coincides with the notion of a proper strict coloring of a C -hypergraph in the context of the recent theory of coloring mixed hypergraphs [15]. In this work instead of studying the upper chromatic number our interest is in determining (or estimating) the balanced upper chromatic number, which arises from balanced rainbow-free colorings. We will define all concepts in the general setting of hypergraphs although here we only study projective planes and projective spaces. Let H be a hypergraph and k be a positive integer. A k-coloring of H is a surjective mapping from V (H ) to a set of k colors; that is c : V (H ) → {0, 1, . . . , k − 1}. The inverse images of the colors are called the color classes; that is Ci = {v ∈ V (H ) : c (v) = i} for i ∈ {0, . . . , k − 1}. Given a k-coloring of H an edge of H is called rainbow, if it is totally multicolored (i.e. the coloring assigns pairwise distinct colors to its vertices). A k-coloring of H is said to be rainbow-free, if it contains no rainbow edges. The upper chromatic number of H, denoted by χ (H ), is the largest integer k for which there is a rainbow-free k-coloring of H. We shall observe that, if χ (H ) = k then k + 1 is the minimum integer with the property that every (k + 1)-coloring of H contains a rainbow edge. In this sense the problem of determining the upper chromatic number is considered an extremal anti-Ramsey problem.

E-mail addresses: [email protected] (G. Araujo-Pardo), [email protected] (Gy. Kiss), [email protected] (A. Montejano). http://dx.doi.org/10.1016/j.disc.2015.06.013 0012-365X/© 2015 Elsevier B.V. All rights reserved.

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

2563

The upper chromatic number has been studied in many different contexts and has been redefined several times under different names (see [2,3,6,7,10,13,14] and references therein). More specifically, results in projective planes appear for example in [1,4,5]. A natural lower bound for the upper chromatic number is given in terms of the 2-transversal number τ2 , which is the minimum cardinality of a set of vertices that intersect each edge in at least two vertices. It is not difficult to see that

χ (H ) ≥ |V (H )| − τ2 (H ) + 1

(1)

holds true for every hypergraph H, and it is an interesting problem to determine when this inequality is tight (see [4]). Generally, the structure of rainbow-free colorings is very specific. Indeed, in order to avoid rainbow edges most of the times very small color classes are needed. For instance, the coloring that provides the above inequality has |V (H )| − τ2 (H ) color classes with one element each and one big color class of order τ2 (H ). Such a coloring is called a trivial coloring [4]. The main motivation of this work is to investigate (in contrast to the latter fact), if there are rainbow-free colorings with color classes of almost the same size. A balanced coloring is a coloring in which the cardinality of all color classes differs at most in one. Hence, balanced colorings are in some sense the opposite of trivial colorings. The balanced upper chromatic number of H, denoted by χ b (H ), is the largest integer k for which there is a balanced rainbow-free k-coloring of H. Obviously χ b (H ) ≤ χ (H ) holds true for any hypergraph H. In this work we are interested in showing that the gap between χ b (H ) and χ (H ) is large for hypergraphs defined by projective planes. In other words, if we forbid trivial colorings then to avoid rainbow-free colorings we are forced to use much fewer colors. How much fewer is the question that we want to answer. A projective plane can be considered as a hypergraph where the set of vertices is the set of points, and the edges are the lines. We denote a finite projective plane of order q by Πq and we will use the same notation to refer to the hypergraph as defined above. So, since Πq has v = q2 + q + 1 points and the same number of lines each containing q + 1 points, then the hypergraph Πq is a (q + 1)-uniform hypergraph of order and size v = q2 + q + 1. It is an easy exercise to verify that the Fano plane Π2 satisfies χ (Π2 ) = 3 and χ b (Π2 ) = 2, while the projective plane of order 3, Π3 , satisfies χ (Π3 ) = 6 and χ b (Π3 ) = 4. For q = ph , being p a large enough prime, it has been recently proved that concerning PG(2, q) the equality holds true in (1), and it is reached only by trivial colorings [4]. In this work we investigate the balanced upper chromatic number χ b (Πq ) of hypergraphs defined by cyclic projective planes. In Section 2 we recall the notion of cyclic planes. Theoretically, the class of cyclic projective planes is wider than the class of Desarguesian planes, but each known finite cyclic plane is isomorphic to PG(2, q) for a suitable q. We use the polygon model in order to prove that: q2 + q + 1 6

≤ χ b (Πq ) ≤

q2 + q + 1 3

if Πq is a cyclic plane of order q. We also show that the upper bound is sharp, if q ≡ 1 (mod 3). We denote the n-dimensional projective space over the finite field of q elements by PG(n, q). In Section 3 we investigate the hypergraphs arising from PG(n, q), n ≥ 3. We prove some bounds on the balanced chromatic numbers, and give constructions of new rainbow-free colorings of these hypergraphs. 2. Balanced coloring in cyclic planes First we give a general upper bound on the balanced upper chromatic number of finite projective planes which is a consequence of an easy counting argument. Theorem 2.1. All balanced rainbow-free colorings of any projective plane of order q satisfy that each color class contains at least three points. Thus

χ b (Πq ) ≤

q2 + q + 1 3

.

Proof. Suppose to the contrary that the plane has a balanced rainbow-free coloring with a color class of size two. For i ∈ {1, 2, 3}, let ni be the number of color classes of size i. Then n2 > 0, and either n1 = 0 or n3 = 0. Since each point of the plane belongs to exactly one color class, then q2 + q + 1 = n1 + 2n2 + 3n3 .

(2)

On the other hand, the number of 2-secant lines of a set of one, two, or three points is respectively equal to zero, one, or at most three. Thus the total number of lines which are not rainbow is n2 if n3 = 0, and at most n2 + 3n3 if n1 = 0. In both cases, according to (2), we obtain less than q2 + q + 1 not rainbow lines, contradicting that the coloring is rainbow-free. 

2564

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

In the rest of this section we consider cyclic projective planes. First of all, we recall the notion of difference sets and some well-known facts about them (Proposition 2.2). We also recall a useful model of cyclic projective planes, due to Kárteszi [12], that we will use throughout this paper. Let G be an additive group. A subset D = {d0 , d1 , . . . , dk } of G is called a difference set if, for every g ∈ G, g ̸= 0, there exists a unique pair of distinct elements di , dj ∈ D such that g = di − dj . The following proposition is an immediate consequence of the previous definition. Proposition 2.2. If D ⊂ G is a difference set and t ∈ G is an arbitrary element, then both −D := {−di : di ∈ D} and D + t := {di + t : di ∈ D} are difference sets.  Let q > 1 be an integer, and v = q2 + q + 1. If the group Zv = Z/v Z contains a difference set D = {d0 , d1 , . . . , dq } then there exists a projective plane called cyclic projective plane of order q, defined as follows: the points are the elements of Zv , 1 that is the set of integers {0, 1, . . . , v − 1}, and the lines are the sets {Li }v− i=0 , where Li := D + i = {d0 + i, d1 + i, . . . , dq + i}. Throughout the paper when we deal with elements of Zv all sums are taken modulo v . The following representation of the cyclic planes comes from Kárteszi [12], and is useful to illustrate our proofs: Consider the numbering of the vertices of a regular v -gon with the elements of Zv in clockwise order. Note that the subpolygon with q + 1 vertices induced by a difference set D has the property that all the chords obtained by joining pairs of their points have different lengths, and it represents the line L0 of Πq . Moreover the line Li is obtained by rotating L0 around the center by angle 2π · vi for i ∈ {1, . . . , v − 1}. In the rest of this section, we present our results related to balanced colorings in cyclic planes. First we give a general lower bound in terms of the smallest nontrivial divisor of v = q2 + q + 1. Theorem 2.3. Let Πq be a cyclic projective plane of order q and let p be the smallest nontrivial divisor of v = q2 + q + 1. Then Πq has a balanced rainbow-free coloring with vp color classes. Thus

χ b (Πq ) ≥

q2 + q + 1 p

.

Proof. For 0 ≤ i ≤ vp − 1 define the color classes as:

 Ci =

i, i +

v p

, . . . , i + (p − 1)

v p



.

Each line of a cyclic projective plane of order q contains a pair of points for which its difference is vp , hence each line contains a pair of points of the form {j, j + vp }. By definition, the two points of every such a set are in the same color class, thus the coloring is rainbow-free.  Note that if v has no ‘‘small’’ nontrivial divisors then the previous bound provides not much information (actually, if v is a prime number then the theorem above is useless). In contrast, if v is divisible by 3, that is p = 3, which is the case when q ≡ 1 (mod 3), then the bound in Theorem 2.3 provides the exact value of the balanced upper chromatic number. Corollary 2.4. If q ≡ 1 (mod 3) then each cyclic plane of order q has a balanced rainbow-free coloring with v3 color classes. Therefore, in this case

χ b (Πq ) =

q2 + q + 1 3

.

Proof. The upper bound is obtained by Theorem 2.1. The lower bound is obtained by Theorem 2.3 since q ≡ 1 (mod 3) implies v is divisible by 3.  A difference set of Zv may contain the subset {0, 1, 3}. If it happens, we can find an optimal balanced coloring as shown in the next proposition. Proposition 2.5. If Zv has a difference set D containing the subset {0, 1, 3}, then the corresponding cyclic plane of order q has a balanced rainbow-free coloring with ⌊ v3 ⌋ color classes. Hence, in this case

χ b (Πq ) =



q2 + q + 1 3



.

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

2565

Fig. 1. The rainbow-free coloring described in the proof of Proposition 2.5 for v = 31.

Proof. By Corollary 2.4 it is enough to consider the case q ̸≡ 1 (mod 3). This implies that v ≡ 1 (mod 3). Let v = 3m + 1, then each x ∈ Zv \ {0} will be expressed as x = 3a + b where 0 ≤ a ≤ m − 1 and 1 ≤ b ≤ 3; and 0 will be written as 0 = 3m + 1 = v . Define the color classes as follows. For 0 ≤ i ≤ m − 2 let Ci = {3i + b : 1 ≤ b ≤ 3}, and let Cm−1 = {3(m − 1) + b : 1 ≤ b ≤ 3} ∪ {0}. This coloring is balanced with m − 1 classes of size three and one color class of size four (see an example for v = 31 in Fig. 1). To verify that the coloring is rainbow-free we show that for every x ∈ Zv , the line Lx has two points in the same color class. If x = 3a + 1 or x = 3a + 2 with 0 ≤ a ≤ m − 1, consider the pair {x, x + 1} ⊂ Lx and note that both elements belong to the color class Ca . If x = 3a + 3 with 0 ≤ a ≤ m − 2 then consider the pair {x + 1, x + 3} ⊂ Lx and note that both elements belong to the color class Ca+1 . If x = 3(m − 1) + 3 then consider the pair {x, x + 1} = {3m, 0} whose elements belong to Cm−1 , and finally if x = 0, then consider the pair {1, 3}, whose elements belong to C0 .  Our last theorem in this section gives a general lower bound on the balanced upper chromatic number of cyclic projective planes. Before state and prove the theorem we give some notation. Let v = 6s + 1 and consider the cyclic group Zv . We will express an element x ∈ Zv \ {0} as 6a + b where 0 ≤ a ≤ s − 1 and 1 ≤ b ≤ 6; and 0 will be written as 0 = 6s + 1 = v . This description is important because for a given element x = 6a + b the value of a will be related with the color class to which it will belong. Theorem 2.6. Each cyclic projective plane of order q has a balanced rainbow-free coloring with at least v6 color classes. Thus

χ b (Πq ) ≥

q2 + q + 1 6

.

Proof. Because of Corollary 2.4 it is enough to prove the statement when q ̸≡ 1 (mod 3). In this case v ≡ 1 (mod 6). Let v = 6s + 1. Let D = {d0 , d1 , . . . , dq } be a difference set of Zv with d0 < d1 < · · · < dq . By Proposition 2.2 we may assume without loss of generality that d0 = 0 and d1 = 1. By definition there uniquely exists j, such that dj+1 − dj = 2. Let d = dj be 1 a shorthand notation. We may also assume that d ≤ v− , otherwise we consider the difference set −D + 1, which contains 2

1 the subset {0, 1, v − d − 1, v − d + 1} and v − d − 1 < v− . If d = 3 then the statement follows from Proposition 2.5 so 2 we also may assume that d > 3. 1 In brief, D is a difference set of Zv containing the subset {0, 1, d, d + 2} where 4 ≤ d ≤ v− . We express d = 6r + j where 2 s 0 ≤ r < 2 and 1 ≤ j ≤ 6. Next we define a coloring of Zv for each one of the following four cases depending on the value of j (the residue modulo 6 of d). Case 1. Let d = 6r + j with j ∈ {2, 3, 4} then

c (6a + b) =



a s−1

for for

0≤a≤s−1 a=s

and 1 ≤ b ≤ 6, and b = 1.

Case 2. Let d = 6r + 1 then

 c (6a + b) =

a + r (mod s) a s−1

for for for

0≤a≤s−1 0≤a≤s−1 a=s

and b = 1, and 2 ≤ b ≤ 6, and b = 1.

2566

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

Fig. 2. The four cases for v = 31.

Case 3. Let d = 6r + 5 then

 a  

a + r + 1 (mod s) c (6a + b) =  a r

for for for for

0≤a≤ 0≤a≤ a= a=

s−2 s−2 s−1 s

and 1 ≤ b ≤ and b= and 1 ≤ b ≤ and b=

for for for for for for

0≤a≤ 0≤a≤ 0≤a≤ a= a= a=

s−2 s−2 s−2 s−1 s−1 s

and 1 ≤ b ≤ 4, and b = 5, and b = 6, and 1 ≤ b ≤ 5, and b = 6, and b = 1.

5, 6, 6, 1.

Case 4. Let d = 6r + 6 then

a    a + r + 1 (mod s)   a + r + 2 (mod s) c (6a + b) = a    r r +1

Fig. 2 depicts the different distribution of the color classes in any case for v = 31. Note that in all cases c is a balanced s-coloring with s − 1 classes of size six and one color class of size seven. Now for each case 1–4, we will prove that the coloring is rainbow-free by exhibiting, for every x ∈ Zv , two points of the line Lx that belong to the same color classes. Recall that, by definition, for every x ∈ Zv we have {x, x + 1, x + d, x + d + 2} ⊂ Lx and that the color classes are defined by Ci = {x : c (x) = i}. In the following tables we indicate for each case and each x ∈ Zv two points of Lx that belong to the same color class. The subscripts of the color classes are taken modulo s. Case 1. d = 6r + j with j ∈ {2, 3, 4} x = 6a + b

Two points in Lx

Color class

0 ≤ a ≤ s − 1 and 1 ≤ b ≤ 5 or a = s − 1 and b = 6

{x, x + 1}

Ca

0 ≤ a ≤ s − 2 and b = 6 or a = s and b = 1

{x + d, x + d + 2}

Ca+r +1

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

2567

Case 2. d = 6r + 1 x = 6a + b

Two points in Lx

Color class

0 ≤ a ≤ s − 1 and 2 ≤ b ≤ 5 or a = s − 1 and b = 6

{x, x + 1}

Ca

0 ≤ a ≤ s − 1 and b = 1

{x, x + d + 2}

Ca+r

0 ≤ a ≤ s − 2 and b = 6

{x + 1, x + d + 2}

Ca+r +1

a = s and b = 1

{x + 1, x + d + 2}

Cr

x = 6a + b

Two points in Lx

Color class

0 ≤ a ≤ s − 1 and 1 ≤ b ≤ 4 or a = s − 1 and b = 5

{x, x + 1}

Ca

0 ≤ a ≤ s − 2 and b = 5 or a = s − 1 and b = 6

{x + 1, x + d}

Ca+r +1

Case 3. d = 6r + 5

0 ≤ a ≤ s − 2 and b = 6

{x, x + d}

Ca+r +1

a = s and b = 1

{x, x + d}

Cr

Case 4. d = 6r + 6. x = 6a + b

Two points in Lx

Color class

0 ≤ a ≤ s − 1 and 1 ≤ b ≤ 3 or a = s − 1 and b = 4

{x, x + 1}

Ca

0 ≤ a ≤ s − 2 and b = 4 or a = s − 1 and b = 5

{x + 1, x + d}

Ca+r +1

0 ≤ a ≤ s − 2 and b = 5 or a = s − 1 and b = 6

{x + 1, x + d + 2} if 0≤a≤s−r −2

Ca+r +2

idem

{ x, x + d } if a≥s−r −1

Ca+r +1

0 ≤ a ≤ s − 2 and b = 6

{ x, x + d + 2 }

Ca+r +2

a = s and b = 1

{ x, x + d + 2 }

Cr +1

It is not difficult to verify the information given in the tables. We show the procedure for one example and the reader can verify the rest using similar arguments: Consider the second row in Case 1. We claim that, if x = 6a + 6 with 0 ≤ a ≤ s − 2 then both x + d and x + d + 2 belong to the color class Ca+r +1 where a + r + 1 is taken modulo s. To see this first consider the case when a ≤ s − r − 2 then x + d = 6a + 6 + 6r + j = 6(a + r + 1) + j and x + d + 2 = 6(a + r + 1) + j + 2 belong to the color class Ca+r +1 because a + r + 1 ≤ s − 1 and 4 ≤ j + 2 ≤ 6. On the other hand, if a + r + 1 ≥ s we have that x + d > v = 6s + 1. Due to the fact that we are calculating in the cyclic group Zv it has the following effect in our description: x + d = 6(a + r + 1) + j − 1

2568

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

and x + d + 2 = 6(a + r + 1) + j + 2 − 1 where a + r + 1 is taken modulo s. Still, both x + d and x + d + 2 belong to the color class Ca+r +1 since 1 ≤ j − 1 < j + 1 ≤ 5.



3. Balanced coloring in higher dimensional spaces In this section we investigate hypergraphs arising from n-dimensional projective spaces where n ≥ 3. The n-dimensional projective space of order q exists if and only if q is a prime power and it is unique up to isomorphism. Let Vn+1 be an (n + 1)-dimensional vector space over the finite field of q elements, GF(q). The n-dimensional projective space PG(n, q) is the geometry whose k-dimensional subspaces for k = 0, 1, . . . , n are the (k + 1)-dimensional subspaces of Vn+1 . For the detailed description of these spaces we refer the reader to [9]. n The basic combinatorial properties of PG(n, q) can be described by the q-nomial coefficients. The expression k q equals to the number of k-dimensional subspaces in an n-dimensional vector space over GF(q), hence it is defined as

n k

:= q

(qn − 1)(qn − q) . . . (qn − qk−1 ) . (qk − 1)(qk − q) . . . (qk − qk−1 )

The proof of the following proposition is straightforward. Proposition 3.1. • The number of k-dimensional subspaces in PG(n, q) is



n +1 k+1

 q

. n+1

• In particular both the number of points and the number of hyperplanes in PG(n, q) equal to q q−1−1 .  −m  • The number of k-dimensional subspaces of PG(n, q) through a given m-dimensional (m ≤ k) subspace in PG (n, q) is nk− . m q A k-spread of PG(n, q) is a set of pairwise disjoint k-dimensional subspaces, giving a partition of the set of points in the geometry. The following theorem gives a necessary and sufficient condition for the existence of spreads. Theorem 3.2 ([8], Theorem 4.1). There exists a k-spread in PG(n, q) if and only if (k + 1)|(n + 1). 1 is the greatest value of k for which k-spreads of PG(n, q) exist. These spreads have an important If n is odd then k = n− 2 property that we will use for constructing balanced rainbow-free colorings. 1 Proposition 3.3. Let S be an n− -spread in PG(n, q). Then each hyperplane of PG(n, q) contains exactly one element of S . 2 1 Proof. If A and B are two n− -dimensional subspaces in a hyperplane H , then the dimension formula gives 2

dim A + dim B = dim⟨A, B⟩ + dim(A ∩ B). Hence dim(A ∩ B) = dim A + dim B − dim⟨A, B⟩ ≥ 2 ·

n−1 2

− dim H = 0,

so A ∩ B is nonempty. But the elements of S are pairwise disjoint, hence each hyperplane contains at most one element of S . 1 On the other hand, by Proposition 3.1 the number of (n − 1)-dimensional subspaces containing a given n− -dimensional 2 subspace is



1 n − n− 2

n−1−

n−1 2

 = q

The spread S consists of



q

n+1 2



n+1 2 n−1 2

qn+1 −1 n+1 q 2 −1

 =

q

q

=q

n+1 2

n+1 2

−1 . q−1

+ 1 subspaces. Each element of S is contained in

n+1 q 2 −1 q −1

hyperplanes. But

 q n+2 1 − 1 qn+1 − 1 +1 · = , q−1 q−1

that is the total number of hyperplanes in PG(n, q). Thus any hyperplane must contain exactly one element of S .



If 0 < d < n are fixed integers, let H (q, n, d) be the hypergraph whose vertex-set is the set of points of PG(n, q) and the edges are the d-dimensional subspaces of PG(n, q). Theorem 3.4. Let n ≥ 3 be an odd number and q be a prime power. Then

χ b (H (q, n, n − 1)) ≥

qn+1 − 1 q−1

−q

n+1 2

− 1.

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

2569 n+1 2

1 Proof. The divisibility condition ( n− + 1)|(n + 1) implies that PG(n, q) has an n−2 1 -spread S which consists of t = q + 1 2 subspaces. We define the coloring in the following way. Take two points from each element of S , these t pairs form t color classes

of size two. Each of the remaining qn+1 −1 q−1

qn+1 −1 q −1

− 2t points forms a color class of size one. Thus the number of the color classes is

n+1 2

−q − 1. It follows from Proposition 3.3 that each hyperplane H of PG(n, q) contains one element of S . This element has the points of a color class of size two, so these two points are contained in H, too. Hence this coloring of H (q, n, n − 1) is rainbowfree.  In the case n = 3 the rainbow-free coloring of Theorem 3.4 is the best possible. Theorem 3.5. Each balanced rainbow-free coloring of H (q, 3, 2) has at most q3 + q color classes. Hence

χ b (H (q, 3, 2)) = q3 + q. Proof. The inequality χ b (H (q, 3, 2)) ≥ q3 + q follows from Theorem 3.4. Suppose that H (q, 3, 2) admits a balanced rainbow-free coloring with c > q3 + q color classes. As PG(3, q) contains q3 + q2 + q + 1 points, this implies that there are at most q2 color classes of size two and all other color classes have size one. If the color class C contains the points P and R, then the number  of  planes of PG(3, q) containing both P and R is the same as the number of planes containing the line PR. This number is

3−1 2−1

= q + 1. Hence the total number of rainbow-free planes

q

is at most q (q + 1) < q + q + q + 1. This contradiction proves the theorem. 2

3

2



The following inductive construction is based on the geometric properties of cones. Theorem 3.6. If H (q, n, d) has a special balanced rainbow-free coloring, namely that each color class has the same size, say k, qn+1 −1

then H (q, n + 1, d) also has a balanced rainbow-free coloring with k(q−1) color classes. Proof. Let Sn be a hyperplane in PG(n + 1, q) and V be a point in PG(n + 1, q) \ Sn . As Sn is isomorphic to PG(n, q), we can qn+1 −1

take a balanced rainbow-free coloring of Sn with t = k(q−1) color classes, C1 , C2 , . . . , Ct such that |Ci | = k if i = 1, 2, . . . , t . Now we define the color classes on the point-set of PG(n + 1, q). For i = 1, 2, . . . , t − 1 let Ci′ = {R ∈ PG(n + 1, q) : (⟨V , R⟩ ∩ Sn ) ∈ Ci } and finally let Ct′ = {R ∈ PG(n + 1, q) : (⟨V , R⟩ ∩ Sn ) ∈ Ct } ∪ {V }. This coloring is well-defined, because if R ∈ PG(n + 1, q) \ {V } is an arbitrary point, then ⟨V , R⟩ is a line ℓ. As V ̸∈ Sn , this line is not contained in Sn , hence the point ℓ ∩ Sn is unique, it belongs to exactly one of the color classes C1 , C2 , . . . , Ct . The coloring has t color classes by definition. The color class Ct′ has size kq + 1 and all other classes have size kq, thus the coloring is balanced. The d-dimensional subspaces containing V are rainbow-free, because each of them contains at least one line passing through V . If a d-dimensional subspace Sd does not contain V then ⟨V , Sd ⟩ is a (d + 1)-dimensional subspace, say Sd+1 . The hyperplane Sn does not contain V , hence Sn ∩ Sd+1 is a d-dimensional subspace in Sn . Thus it contains at least two points, say X and Y from a color class Ci . This implies that the points ⟨V , X ⟩ ∩ Sd and ⟨V , Y ⟩ ∩ Sd are in Ci′ , hence Sd also contains two points from the same color class, so the coloring is rainbow-free.  Corollary 3.7. Let p be the smallest nontrivial divisor of v = q2 + q + 1. Then H (q, 3, 1) has a balanced rainbow-free coloring with vp color classes. In particular if q ≡ 1 (mod 3) then

χ b (H (q, 3, 1)) ≥

q2 + q + 1 3

.

Proof. The projective plane PG(2, q) is cyclic, thus the construction given in Theorem 2.3 implies that H (q, 2, 1) has a balanced rainbow-free coloring with vp color classes and each class has size p.  In the case q ≡ 1(mod 3) the following upper bound on χ b (H (q, 3, 1)) is close to the lower bound given by Corollary 3.7. Theorem 3.8. The size of the larger color classes in a balanced rainbow-free coloring of the points with respect to the lines in PG(3, q) is at least 2q + 2, hence

χ b (H (q, 3, 1)) ≤

q2 + 1 2

.

2570

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

Proof. Suppose that a balanced rainbow-free coloring contains nk−1 color classes of size k − 1, and nk ̸= 0 color classes of size k. Then q3 + q2 + q + 1 = (k − 1)nk−1 + knk . The total number of lines in PG(3, q) is (q2 + 1)(q2 + q + 1) and each line must contain at least two points from the same color class, consequently

 nk−1 ·

k−1



  k

+ nk ·

2

2

≥ (q2 + 1)(q2 + q + 1).

But,

 nk−1 ·

k−1



  k

+ nk ·

2

2

≤ ((k − 1)nk−1 + knk )

k−1 2

= (q3 + q2 + q + 1)

k−1 2

.

Hence,

(q3 + q2 + q + 1)(k − 1) ≥ 2(q2 + 1)(q2 + q + 1). From this inequality we obtain k − 1 ≥ 2q +

2 q+1

,

but k is an integer, hence k ≥ 2q + 2.



4. Final comments First of all we pointed out the obtained gap between the balanced upper chromatic number, χ b (H ), and the upper chromatic number, χ (H ), when considering hypergraphs defined by projective planes. As mentioned in the introduction, it has been shown by Bacsó, Héger, and Szőnyi that there are projective planes for which

χ (Πq ) = v − τ2 (Πq ) + 1,

(3)

where v = q2 + q + 1 is the number of points and lines in Πq . Specifically, they proved that for PG(2, q) where q = ph and p is a prime with, p ≥ 29 and h ≥ 3, or q a square such that√q > 256, the equality in (3) holds true. It is well known that in such cases τ2 (Πq ) ≈ c1 q and therefore χ (Πq ) ≈ v − c1 v where c1 ≈ 2.5, (see [4]). While concerning the balanced upper chromatic number we obtain χ (Πq ) ≈ c2 v with 61 ≤ c2 ≤ 13 . In Corollary 2.4 we obtain χ (Πq ) = v3 when q ≡ 1 (mod 3) and in Proposition 2.5 we prove for q̸≡ 3 (mod  3) and the difference set of the corresponding cyclic plane contains q2 +q+1 3

the subset {0, 1, 3} we have that χ b (Πq ) =

coloring we conjecture that the equality χ (Πq ) =



. Although in Theorem 2.6 we are only able to give a v6 rainbow-free

q2 +q+1 3



holds true in all the cases.

On the other hand with a slight modification of the proof of Theorem 2.6 we can define a balanced rainbow-free coloring v of each cyclic plane with around 20 color classes, such that it has the advantage that one can change the color of some points (at least one in each color class) freely without violating the rainbow-free property. If we combine this coloring and the cone construction given in Theorem 3.6 using the ‘‘free’’ points in the plane, we can balance the sizes of the color classes in the 3-space and we can prove that χ b (H (q, 3, 1)) ≥ q 2 +1 . 2

χ b (H (q, 3, 1)) ≤

q2 . 20

This lower bound is far away from the general upper bound

We conjecture that Corollary 3.7 gives the best construction and χ b (H (q, 3, 1)) ≤ q 2 +q +1

q 2 +q +1 3

holds for

all q, and the general lower bound is somewhere around . 6 We also guess that the spreads give the best possible colorings, namely equality holds in Theorem 3.4 for all n, hence

χ b (H (q, n, n − 1)) =

qn+1 −1 q−1

n+1 2

− 1. We were not able to prove this by usual double counting, perhaps one can do so by applying some sophisticated techniques. By the previous analysis we conclude this work with the following two conjectures: −q

Conjecture 4.1. Each cyclic projective plane of order q satisfies the equality:

χ b (Πq ) =



q2 + q + 1 3



.

Conjecture 4.2. The 3-dimensional projective space of order q satisfies the inequalities: q2 + q + 1 6

≤ χ b (H (q, 3, 1)) ≤

q2 + q + 1 3

.

G. Araujo-Pardo et al. / Discrete Mathematics 338 (2015) 2562–2571

2571

Acknowledgments We would like to thank the Editor and the anonymous reviewers for their insightful comments and suggestions to this paper. Third author’s research was supported by the Mexican–Hungarian Intergovernmental Scientific and Technological Cooperation Project, Grant No. TÉT 10-1-2011-0471, the Hungarian National Foundation for Scientific Research, Grant No. K 81310, CONACyT-México under Projects 166306, 178395, 219827 and PAPIIT-México under Projects IN101912, IN104915, IN112614, IA102013. References [1] M.G. Araujo Pardo, Daisy structure in Desarguesian projective planes, J. Aust. Math. Soc. 74 (2003) 145–153. [2] J.L. Arocha, A. Montejano, Null and non-rainbow colorings of projective plane and sphere triangulations, Discrete Appl. Math. (2015) in press, http://dx.doi.org/10.1016/j.dam.2015.04.007. [3] M. Axenovich, D. Fon-Der-Flaass, On rainbow arithmetic progressions, Electron. J. Combin. (2004) 11:R1. [4] G. Bacsó, T. Héger, T. Szőnyi, The 2-blocking number and the upper chromatic number of PG(2, q), J. Combin. Des. 21 (2013) 585–602. [5] G. Bacsó, Zs. Tuza, Upper chromatic number of finite projective planes, J. Combin. Des. 7 (2007) 39–53. [6] P. Erdős, M. Simonovits, V. Sós, Anti-Ramsey theorems, in: Infinite and Finite Sets, in: Discrete Math., vol. 10, Colloq. Math. Soc. Janos Bolyai, 1975, pp. 633–643. [7] S. Fujita, C. Magnant, K. Ozeki, Rainbow generalizations of Ramsey theory: A survey, Graphs Combin. 26 (1) (2010) 1–30. [8] J.W.P. Hirschfeld, Projective Geometries over Finite Fields, second ed., Clarendon Press, Oxford, 1998. [9] J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991. [10] V. Jungić, D. Král, R. Škrekovski, Coloring of plane graphs with no rainbow faces, Combinatorica 26 (2006) 169–182. [11] V. Jungić, J. Nešetřil, R. Radoičić, Rainbow Ramsey Theory, Integers 5 (2) (2005) A9. [12] F. Kárteszi, Introduction to Finite Geometries, North-Holland Publishing Co., Amsterdam-Oxford, 1976. [13] D. Král, Mixed hypergraphs and other coloring problems, Discrete Math. 307 (2007) 923–938. [14] A. Montejano, O. Serra, Rainbow-free 3-colorings in Abelian groups, Electron. J. Combin. 19 (2012) #P45. [15] V.I. Voloshin, Coloring Mixed Hypergraphs: Theory, Algorithms and Applications, Amer. Math. Soc. Providence, 2002.