On the upper chromatic number and multiplte blocking sets of PG($n,q$)

We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small $t$-fold (weighted) $(n-k)$-blocking set of $\mathrm{PG}(n,p)$, $p$ prime, must contain the weighted sum of $t$ not necessarily distinct $(n-k)$-spaces.

and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for ≤ t p + 1 3 8 , a small t-fold (weighted) n k ( − )-blocking set of n p p PG( , ), prime, must contain the weighted sum of t not necessarily distinct n k ( − )-spaces.

| INTRODUCTION AND RESULTS
Throughout the paper, let denote a hypergraph with point set V and edge set E. A strict N -coloring of is a coloring of the elements of V using exactly N colors; in other words, ). Given a coloring , we define the mapping → φ V N : {1,2,…, }by φ P i ( ) = if and only if ∈ P C i . We call the numbers N 1, …, colors and the sets C C , … N 1 color classes. We call a hyperedge ∈ H E rainbow (with respect to ) if no two points of H have the same color; that is, . The upper chromatic number (or shortly UCN) of the hypergraph , denoted by χ̄( ), is the maximal number N for which admits a strict N -coloring without rainbow hyperedges. Let us call such a coloring proper or rainbow-free. It is easy to see that for an ordinary graph G (ie, a 2-uniform hypergraph), χ Ḡ ( ) is just the number of connected components of G. As one can see, the above defined hypergraph coloring problem is a counterpart of the traditional one, where we seek the least number of colors with which we can color the vertices of a hypergraph while forbidding hyperedges to contain two vertices of the same color. The general mixed hypergraph model, introduced by Voloshin [13,14], combines the above two concepts. This mixed model is better known but here we do not discuss it; the interested reader is referred to [8,15].
It is clear that if we find a vertex set ⊂ T V in which intersects every hyperedge in at least two points, then by coloring the points of T with one color and all the other points of V by mutually distinct colors, we obtain a proper, strict V T (| | − | | + 1)-coloring.
The size of the smallest t-transversal of is denoted by τ ( ) t . Definition 1.2. We say that a coloring of is trivial if it contains a monochromatic 2-transversal.
As seen above, the best trivial colorings immediately yield a lower bound for χ̄( ).  Two general problems are to determine whether this bound is sharp (for a particular class of hypergraphs) and to describe the colorings attaining the upper chromatic number. In this paper, the hypergraphs we consider consist of the points of the n-dimensional projective space n q PG( , ) over the finite field q GF( ) of q elements with its k-dimensional subspaces as hyperedges, ≥ ≤ ≤ n k n 2, 1 − 1. We denote this hypergraph by n k q ( , , ); however, we usually take into account the richer structure of n q PG( , ) when working in n k q ( , , ). The study of this particular case was started in the mid-nineties by Bacsó and Tuza [3], who established general bounds for the upper chromatic number of arbitrary finite projective planes (considered as a hypergraph whose points and hyperedges are the points and lines of the plane). Let us introduce the notation ⋯ θ θ q q q for the number of points in a k-dimensional projective space of order q. We recall that a projective plane of order q has θ q q = + + 1 2 2 points.
In this study, we determine χ n k q ( ( , , )) for many parameters and aim not only to characterize trivial colorings as the only ones achieving the upper chromatic number of the hypergraph n k q ( , , ), but to obtain results showing that proper colorings of n k q ( , , ) using a little less number of colors than χ n k q ( ( , , )) are trivial; in other words, to prove that trivial colorings are stable regarding the number of colors. For the sake of convenience, we will formulate our results in three theorems for the hypergraph n n k q ( , − , ). Let us note that if k < n 2 , then τ n n k q θ ( ( , − , )) = 2 k 2 , where equality can be reached by the union of two disjoint k-spaces, but not much is known if ≥ k n 2 ; some details are given in Section 2.
Under these assumptions the following hold: a) If k < n 2 , then any rainbow-free coloring of n n k q ( , − , ) using colors contains a monochromatic pair of disjoint k-spaces, and hence is trivial. b) If ≥ k n 2 , then Note that the stability gap δ in the above result is far much weaker in the nonprime case (in particular, the case k = 1 is missing). The next theorem gives a much better result at the expense of requiring much stronger assumptions on the order and the characteristic of the field.
Then any rainbow-free coloring of n n k q ( , − , ) using colors contains a monochromatic 2-transversal, and hence is trivial.
The requirements on q and N in the above theorem could be chosen differently, see Remark 3.19 for the details. Note that Theorem 1.8 is not phrased in terms of τ n n k q ( ( , − , )) 2 , the parameter found in the trivial lower bound Proposition 1.3. As noted earlier, τ n n k q θ ( ( , − , )) = 2 k 2 if k < n 2 . If k = n 2 , then [9] asserts the existence of a 2-fold k-blocking set in k q PG(2 , ) (in finite geometrical language, t-transversals of n n k q ( , − , ) are called t-fold k-blocking sets, see Section 2) of size q 2 + 2 , whence Theorem 1.8 yields that the trivial bound is again sharp for k k q (2 , , ), regardless the exact value of τ k k q ( (2 , , )) 2 .
In the proof of the above theorem, we rely on weighted 2-fold blocking sets as well, so we devote the next section to this topic, and we obtain the following new result which, in fact, follows from the similar Theorem 2.9 about t p (mod ) sets. The precise definitions are given in the next section. Theorem 1.9. Let be a minimal weighted t-fold k-blocking set of n p p PG( , ), prime. Assume that ≤ t p | | ( + ) − k 1 2 1 2 and ≤ t p + 1 3 8 . Then is the (weighted) union of t not necessarily distinct k-dimensional subspaces.

| SMALL, WEIGHTED MULTIPLE n k ( − )-BLOCKING SETS
For the sake of convenience, we will refer to n k ( − )-blocking sets instead of k-blocking sets throughout this section.

| Preliminary notation and results
Definition 2.1. An m-space is a subspace of n q PG( , ) of dimension m (in projective sense). A point set in n q PG( , ) is called a t-fold n k ( − )-blocking set if every k-space intersects in at least t points. A point P of is essential if P \{ } is not a t-fold n k ( − )blocking set; in other words, if there is a k-space through P that intersects in precisely t points. is called minimal, if all of its points are essential; in other words, if does not contain a smaller t-fold n k ( − )-blocking set.
In n q PG( , ), every n k ( − )-space intersects every k-space nontrivially. If n k − < n 2 , it is easy to find two (or more, say, t) disjoint n k ( − )-spaces, whose union is clearly a 2-fold (or t-fold) n k ( − )blocking set of size θ 2 n k − . If ≥ n k − n 2 , this does not work and, in fact, not much is known even about the size of a smallest double n k ( − )-blocking set, let alone its structure. Even for the particular case n k = 2 , the first and, so far, only general construction for small double n k ( − )blocking sets appeared in [9]. Note that, however, weighted t-fold blocking sets can be obtained easily in this way.
Definition 2.2. A weighted point set of n q PG( , ) is a multiset of the points of n q PG( , ). We may refer to the multiplicities of the points of via a function w w = mapping the point set of n q PG( , ) to the set of nonnegative integers, where w is also called a weight function; points not contained in have weight zero by w and, vice versa, zero weight points are considered to be not in . We call a weighted point set of n q PG( , ) a weighted t-fold n k ( − )-blocking set if for every k-space , and is called minimal if decreasing the weight of any point results in a k-space violating the previous property; in other words, if does not contain a strictly smaller t-fold n k ( − )-blocking set, where the size of a weighted point set is defined as the sum of weights in it. Also, for any point set ∩ S S , | | is defined as ∑ ∈ w P ( )

P S
, and in general, any quantity referring to a number of points of is usually considered with multiplicities. For example, an i-secant line ℓ (with respect to ) is a line such that ℓ ∩ i | | = .
We also refer to 1-fold and 2-fold blocking sets as blocking sets and double blocking sets, respectively; the term multiple blocking set refers to a t-fold blocking set with ≥ t 2. We call a point of weight one simple. It is easy to see that a weighted t-fold k-blocking set must contain at least tθ k points unless ≥ t q + 1. We include this supposedly folklore result with proof for the sake of completeness.
Proof. We prove by induction on k. If k = 1, we may take a point ∉ P (otherwise ≥ θ qθ | | > n n −1 and there is nothing to prove). There are θ n−1 lines through P, each containing at least t points of , whence Note that ≤ t q is necessary here, as if contains each point of an n k ( − + 1)-space with weight one, then is a q ( + 1)-fold n k ( − )-blocking set of size θ qθ q θ = +1<( +1 ) n k n k n k − +1 − − ; moreover, adding s further n k ( − )-spaces to we obtain a weighted q s ( + 1 + )-fold n k ( − )blocking set of size less than q s θ ( + 1 + ) n k − for any ≥ s 0. A stability result for weighted t-fold n k ( − )-blocking sets of size close to this lower bound was proven by Klein and Metsch [12], Theorem 11].
Result 2.4 (Klein and Metsch [12]). Let be a weighted t-fold n k ( − )-blocking set in n q PG( , ). Suppose that ≤ tθ rθ | | + n k n k − −− 2 , where t and r satisfy the following: 0is an integer; c) any blocking set of q PG(2, ) of size at most q t + contains a line.
Then contains the (weighted) union of t not necessarily distinct n k ( − )-spaces.
Let us remark that for k = 1 (ie, when is a t-fold weighted blocking set with respect to lines), [12], Theorem 7] shows that condition c) can be omitted in the above result. However, a blocking set of q PG(2, ) not containing a line must contain at least q q + + 1 points in general (see [7] by Bruen), and, according to the following result of Blokhuis, at least q ( + 1) 3 2 if q is prime, hence condition c) holds accordingly.
The following two theorems will be very useful for us.
Result 2.6 (Harrach [11]). Suppose that a weighted t-fold k-blocking set in n q PG( , ) has less than t q θ ( + 1) + k k−1 points. Then contains a unique minimal weighted t-fold k-blocking set ′.
A theorem of the below type is often called a t p (mod ) result.
Result 2.7 (Ferret, Storme, Sziklai, and Weiner [10]). Let be a minimal weighted t-fold n k ( − )-blocking set of n q q p p PG( , ), = , Finally, we recall that the number of k ( + 1)-spaces containing a fixed k-space in n q PG( , ) is θ n k − −1 . This can be seen easily by taking an n k ( − − 1)-space disjoint from the fixed k-space and observing that each appropriate k ( + 1)-space intersects it in a unique point.

| Proof of Theorem 1.9
We prove a theorem closely related to Theorem 1.9 by considering an analogous problem in a slightly more general setting.
Clearly, t p (mod ) sets are t-fold blocking sets if t p < and, by Result 2.7, small minimal t-fold blocking sets are t p (mod ) sets. Theorem 2.9. Let be a t p (mod ) set with respect to the k-dimensional subspaces in n p p PG( , ), prime. Suppose that ≤ t p + 1 Case 1. k = 1 (and n ≥ 2). Notice first that every point of has weight at most t. Indeed, by taking the weights of the points modulo p, we may assume that no point has weight at least p; and if ≤ ≤ t w P p + 1 ( ) − 1 for a point P, then all the θ n−1 lines through P must contain at least p t w P It follows from Results 2.4 and 2.5 that the assertion holds if tθ | | = n−1 , hence we may assume that tθ | | > n−1 and prove by contradiction. We will call lines that are neither t-secants (to ), nor contained fully in long lines; lines contained in will be referred to as full lines. Non-t-secant lines are, therefore, either full or long. Long lines exist as on any point not in (an outer point) we find a line intersecting in more than t points, since tθ | | = n−1 would follow otherwise. Suppose that the minimum weight of a long line is sp t + . Clearly, ≤ ≤ s t 1 − 1 (the weight of a long line is at most tp). Let ℓ be a long line of weight sp t + , and let ∈ ℓ P \ . We want to show that for any 2-space Π containing ℓ, there is a long line through P in Π different from ℓ. Fix such a plane Π (if n = 2, then this is unique) and suppose to the contrary. Let ∩ ′ = Π. Then, looking around from P in p t sp Π, | ′| = ( + 1) + . Similarly as before, there must be a non-t-secant line on any point ∈ R Π; in other words, long and full lines form a blocking set in the dual plane of Π. It follows that long lines cover each outer point of Π exactly once. Moreover, the number of non-t-secant lines must be at least p ( + 1) 3 2 for the following reason.
By Blokhuis' Result 2.5, a blocking set of p PG(2, ) of size less than p ( + 1) 3 2 contains a line. In our setting this situation would result in a point Q through which all lines are either long or full. But then . Let e be a t-secant to ′ (such a line exists as seen above). Let P P , …, r denote the number of full and long lines on P i , respectively, and let h 1 and h 2 be the total number of full and long lines, respectively; then Looking around from P i we see that be the number of long lines intersecting e in a point of ′.
, and we obtain that As t p < ( + 1) 1 2 , it follows that ≥ s 2 and h p > ( + 1) full lines one by one. The first line contains at least sp t + weights of ′. The second line may intersect it in a point of weight at most t, hence we see at least sp more weights on it. Turning to the full lines, the ith full line contains at least p i p i + 1 − 2 − ( − 1) = − points of ′ not contained by any of the previous lines. Altogether we obtain a contradiction. Thus we see that all planes containing ℓ indeed contain at least one other long line through P, so we find at least θ 1 + n−2 long lines through P, hence on all the θ n−1 lines through P we find that . Project the points of from P into an arbitrary hyperplane H . We get a weighted point set ⊆ Π for which be the k-space spanned by P and W . Then The steps of the proof have a lot in common with those in [2]. We recall that we want to color the points of n q PG( , ) with as many colors as possible so that each n k ( − )-space contains two equicolored points. For two points P and Q PQ , denotes the line joining them.

| 125
Note that Let us collect some facts regarding the above defined q-binomial coefficients.
Proof. The first statement is trivial. As for the second one, let U be the given k-space. The quantity in question is just the number of m k as ≥ s 1 and ≥ q 2. □ General notation and assumptions. Suppose that a strict proper coloring of n n k q ( , − , ) using N colors is given. We denote the color classes of by C C , …, N 1 . For the sake of simplicity, we will compare N with θ θ which, in principle, may be negative as well. Note that the number δ in Theorems 1.7 and 1.8 is an upper bound on d. Without loss of generality we may assume that C C , …, m 1 are precisely the color classes of size at least two for some ≥ m 1.
Definition 3.3. We say that a color class C colors the n k As every n k ( − )-space must be colored by at least one of the color classes among C C , …, m 1 , we clearly see that is a 2-fold k-blocking set.
, | | 2 . This and the previous equality imply Proof. If C colors an n k ( − )-space U , then U contains a line spanned by the points of C.
The number of such lines is at most ( ) C | | 2 . By Lemma 3.2, the number of n k ( − ) spaces containing a given line is The next proposition says that cannot be too large; roughly speaking, Proof. As every n k ( − )-space must be colored, by Lemma 3.5 and convexity we have By Lemma 3.2d) and Proposition 3.4a), b) (here we use Lemma 3.2a)). Then by the assumption and Lemma 3.2 c), the right-hand-side of the above expression is at most a contradiction. □ The following lemma will be very useful as it provides us large color classes if is not large. The proof is based on Result 2.6. Right now, we do not need the following stronger version of this lemma since our blocking set has no weights, but respecting its future use we will state it in a more general setting. This version can deal with colorings which come from weighted blocking sets. BLÁZSIK ET AL. | 127 Lemma 3.7. Suppose that a color class C contains a simple essential point P of . Then there exists a set of simple points ⊂ S C P \{ } such that , and for any point ∈ Q S there exists an n k ( − ) space U such that ∩ U P Q = { ; } (so these points are essential for ). In particular, , let P Q iff Q is also a simple point and there exists an n k ( − )space U such that ∩ U P Q = { , }. As P is simple and essential, we find at least one such point. Let Proof. By Result 2.6 there is a unique minimal 2-fold k-blocking set ′ contained in . By Lemma 3.7 we know that if a color class contains a point of ′, then it contains at least points of it, while all other color classes in have at least two points. This and Proposition 3.4 a) imply that The left-hand side expression is concave in | |. Substituting either which, due to simple calculations and rearrangement, leads to Using these results, we may assume that is quite small. As shown by the next proposition, this immediately gives the desired result on the upper chromatic number provided that contains the union of two disjoint one-fold blocking sets, which property can be deduced from a stability type result on multiple blocking sets like Theorem 1.9 or Result 2.4; however, the strength of the result obtained in this way will be utterly dependent on the strength of the stability result. Proof. We may assume that q θ | | < 3 + k k−1 , otherwise the assertions are trivial. Then, by Result 2.6, ∪ U U ′ = 1 2 is precisely the set of essential points of . If the coloring is not trivial, then there are at least two colors used in ′, say, red and green. Without loss of generality, we may take a red point ∈ P U 1 . By Lemma 3.7, we find a set S of essential points of such that S q θ | | = 3 − | | + k k−1 , and for each point ∈ Q S there is an , so ⊂ S U 2 . By interchanging the role of U 1 and U 2 , we see that U 1 and U 2 both contain at least q θ 3 − | | + k k−1 red points. As the same holds for green points as well, we find that The next lemma shows under what conditions does Proposition 3.6 provide a good enough bound on | | to make Proposition 3.8 work with β = 5, the value we will typically use.
For k = 1, (1) and (3)  Regarding (3), we can similarly deduce that It is easy to see that the latter requirement is weaker for ≥ q 9, so the former one is enough, which is positive if ≥ q 13. Thus under these conditions Proposition 3.8 yields θ d | | < 2 + 2 + 3 k . As the quantities on both sides are integers, the proof is finished. □  . Equality can be reached by trivial colorings since, as k < n 2 , we can always find two disjoint k-spaces, whose union is clearly a 2-fold k-blocking set. □ and ≥ q q 11, prime. As , we can apply Proposition 3.6 with α = 1 4 to obtain showing that no proper coloring satisfies the condition on d.) As q | | < 2.5 k , Proposition 3.9 claims that our coloring is trivial.
Suppose now that q is not a prime, and recall that our assumptions in this case are , so we may apply Result 2.4 with t = 2 and r q = − 2 to see that contains the union of two disjoint k-spaces (again, ≥ k n 2 gives a contradiction). As θ q q | | < 2 + < 2.5 k k k −1 clearly holds, Proposition 3.9 claims that the coloring is trivial. □ Remark 3.12. We do not believe that the upper bound ≲ d q 0.2 k for the q prime case in the above result is close to be sharp. We think that the limit should be roughly ≲ d q 0.5 k but to achieve this, one needs to improve Propositions 3.6 and 3.8 significantly, or to use a different approach. Improving only Proposition 3.6 would allow us to prove the same assertion under ≲ d q 0.25 k (this is the best allowed by Proposition 3.8).
3.2 | Improvements when q is not a prime: The proof of Theorem 1.8 We recall that = ( ) denotes the union of color classes in the proper coloring with at least two elements, so is a 2-fold k-blocking set in n q PG( , ) colored in a way that each n k ( − )space contains at least two points of of the same color. We will rely on some initial assumptions that are essential in the sense that they cannot be changed significantly so that the reasoning still works, which will be derived from the much more restrictive but adjustable requirements of Theorem 1.8 that may be fine-tuned to obtain a similar result. We aim to treat these somewhat separately to make future parameter adjustments easier.
Let ′ denote the unique minimal 2-fold k-blocking set contained in (which is the set of essential points for , cf. Result 2.6). We want to prove that ′ is monochromatic; to this end, let us suppose to the contrary that contains a red and a green essential point as well. As 2 clearly follows, the t p mod property (Result 2.7) holds for ′. We want to show that the coloring is trivial; in other words, contains a monochromatic 2fold k-blocking set.
We consider three cases depending on the relation between n and k 2 . Our main case is when n k = 2 , in which situation the famous André-Bruck-Bose representation of projective planes shall be used to enable us using planar tools. The cases n k > 2 and n k < 2 will be traced back to this one in the following way. During this procedure the dimension of the host space, the dimension of the subspaces we want to color properly, the coloring, etc., may change. We will refer to these modified objects by their original notation equipped with a bar.

| n ≤ 2k
If ≤ n k 2 then we simply embed this projective space into k q PG(2 , ), and let n k = 2 and k k = (ie, we do nothing if n k = 2 ). Color the new points with new and pairwise different colors. After the embedding we get a strict proper coloring of k k q n n k q (2 , , ) = ( , − , ) (a k-space of BLÁZSIK ET AL. | 131 k q PG(2 , ) intersects the embedded n-space containing in a k n k n k + − 2 = − dimensional subspace). Note that and ′ are left unchanged, so (this last upper bound stands here for future purposes).

> 2
Let us embed n q PG( , ) into n k q PG(2 − 2 , ) and let us take an n k ( − 2 − 1)-space ⊂ n k q PG(2 − 2 , ) which is disjoint from n q PG( , ) (considered now as a given n-space of n k q PG(2 − 2 , )); thus n k q PG(2 − 2 , ) is generated by the original n q PG( , ) and V. We build a cone upon the base with vertex V; that is, the cone consists of the points of the lines joining a point ∈ X with a point V ∈ Y .
Lemma 3.13. For an arbitrary point ) there exist a unique pair of points ∈ X n q PG( , ) and V ∈ Y such that the line defined by X and Y contains P.
Proof. If a good pair X Y , exists then, clearly, the line XY is contained in ,P G ( , ), which is a subspace of dimension n k ( − 2 − 1 + 1) + n n k ( + 1) − (2 − 2 ) = 1. Hence a line of this type is unique, and it defines the points X and Y in a unique way. □ The points of n k q PG(2 − 2 , ) not in get pairwise distinct new colors, and let us color the points of in the following way. The points of V will get the color of an arbitrarily chosen point of , and the points of V ∪ \( ) get the color of their well-defined ancestor (the unique point X in Lemma 3.13) in . Finally, let us give weight two to the points of V. In this way, the coloring of n k n k q (2 − 2 , − , ) is proper, since if an n k ( − )-space U meets V then it is blocked by trivially, and if it is skew to V then V 〈 〉 U , will be an n k (2 − 3 )-space such that it meets n q PG( , ) in an n k ( − )-space W , thus W contains two points of of the same color and, by the cone structure, U contains two points of of the same color. Also, the red and the green essential points for in n q PG( , ) remain essential for , hence contains a red and a green essential point of weight one.
Let n n k k n k = 2 − 2 , = − . Note that except for the points of V, each point of has weight one, and . Furthermore, the number of points of (with weights) is ; on the other hand, from the assumption

= 2
In both of the above cases, we ended up in a projective space of order n k = 2 whose points admit a proper coloring with respect to k -spaces, and the union of the color classes of size at least two form a 2-fold k -blocking set of size at most ≤ θ q | | 2.2 − 4 k k −1 on the one hand, and ≤ q γq | | 2( + 1) + k k on the other hand. Moreover, is either nonweighted, or the set of points with weight more than one is a subplane of dimension at most k − 2, and all points in this subplane are of weight two. In both cases, our indirect assumption assures that there exist red and green essential points of weight one. From now on we will work in this setting only, so we reset the notation and omit the bars.
For future purposes, we need to find a hyperplane H that intersects in at most θ 2.2 k−1 points and contains all points of weight two (if there is any). If k = 1, we are done (otherwise blocks every line of q PG(2, ) at least three times, so ≥ q | | 3( + 1), a contradiction). Suppose now ≥ k 2, and recall -space consisting of the points of weight two or, if there are no such points, an arbitrary k ( − 2)-space. Among the θ k+1 distinct k ( − 1)-spaces containing U −2 , there must be one, say, U −1 , that contains no point of , a contradiction. Among the θ k distinct k-spaces containing U −1 there must be one, say, U 0 , that contains at most two points of . Then among the θ k i − −1 distinct k i ( + + 1)-spaces containing U i , there must be one, say, U i+1 , that contains at most , a contradiction. To find an appropriate hyperplane U k−1 , we claim that among the θ , a contradiction (where we use ≥ q 25). Thus we find an n We set H U = k−1 to be the hyperplane (a k (2 − 1)-space) admitting the properties claimed. André [1] and independently Bruck and Bose [5,6] developed a method, the well-known André-Bruck-Bose representation, for representing translation planes of order q h with kernel containing q GF( ) in the projective space h q PG(2 , ). It arises from a suitable h ( − 1)-spread of the hyperplane at infinity in h q PG(2 , ). The affine lines of the plane are h-dimensional subspaces containing the h ( − 1)-spaces of the h ( − 1)-spread. The ideal points correspond to the elements of the spread. Thus a point set intersecting every h-space yields a blocking set in the plane q PG (2, ) h .
It is well-known that an arbitrary k ( − 1)-space can be mapped to any other k ( − 1)-space with a suitable linear transformation. By the previous observations we can take a k ( − 1)-spread of H (ie, a set of k ( − 1)-spaces that partition H ) in such a way that if V exist it will be contained in one of the spread elements. Moreover, this transitivity property allows us to choose such a Desarguesian (also called regular) spread, too.
Remember that we have already assumed on the contrary that ′, the minimal part of , contains red and green essential points of weight one. By using Lemma 3.7 and the choice of H one can see that ′ must have both red and green affine points. In the following we will show that the minimal part of must be monochromatic which will give us a contradiction.
Let us define a point-line incidence structure H Π = Π( , ) in the following way: • the points of Π are the points of k q H PG(2 , ) \ and the elements of ; } is considered to be a line of Π, as well as ; • a point is incident with a line if it is an element of it.
Then Π is well-known to be a projective plane of order ≔ ͠ q q k by the André-Bruck-Bose representation, and since is a Desarguesian spread, then ≃ ͠ q Π PG(2, ).
We will consider as the line at infinity in Π, and a point of Π is called ideal or affine according to whether it is on the ideal line or not.
Definition 3.14. From the coloring of k q PG(2 , ), we define a coloring ∼ of the points of Π in the following way.
• For an affine point P of Π, let P inherit its color naturally from the coloring .
• For an ideal point ∈ S , we distinguish two cases. On the one hand, if each point of S forms a singleton color class of (ie, ∩ ∅ S = ), then let the color of S be the color of an arbitrarily chosen point of S. On the other hand, if there is a color class of of size at least two containing a point of S (ie, ∩ ≠ ∅ S ), then color S with a color i such that Note that ∼ is an N -coloring of Π that might not be strict.
Definition 3.15. From a weighted point set X of k q PG(2 , ) with weight function w X , we define a weight function ∼ w X on the points of Π in the following way.
• For an affine point P of Π, let ∼ w P w P For the point set ∼ X X , denotes the weighted point set of Π corresponding to the weight function ∼ w X (zero weight points are not considered as elements of ∼ X ).
Consider now ∼ ∼ = ( ) (recall that may be weighted). If ∉ ∼ S , then the points of S have pairwise distinct colors in (and all are singletons). If ∈ ∼ S is of weight one, then the color of S at ∼ is the same as the color of the unique point in ∩ S at . We remark that for the union ∼ ( ) of color classes of size at least two of ∼ in ⊆ ∼ ∼ Π, ( ) , but equality does not follow immediately from the definitions. In the sequel, we will work with ∼ using the property that every line of Π intersects it in at least two equicolored points, yet we make the following, slightly stronger observation.
Proposition 3.16. The coloring ∼ with the weight function ∼ w is a proper weighted coloring of Π; that is, every line of Π contains some points of the same color whose weights add up to at least two.
Proof. Let U be the k-space of k q PG(2 , ) corresponding to a line ℓ of Π, and let follows), then, as is proper, U contains two distinct points of the same color with respect to , say, P and Q; note that ⊂ P Q { , } . If both P and Q are affine points (which is the case if ∼ w S ( ) = 0), then we are also done. Suppose now ∈ ∉ P S Q S , and ∼ w S ( ) = 1. Then, as P is the unique point of ∩ S , the color of S at ∼ is the same as the color of P at , and so S If there are other color classes in containing more than two points, replace their color by red. In this way we obtain a nontrivial proper coloring ′ such that ∼ ∼ ( ) = ( ′); thus it is enough to restrict our attention for colorings admitting only two color classes of size more than one. Then the points of  are also either red or green and both colors actually occur in the affine part. Denote the set of red points of  by  r and the set of green ones by  g .

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Let us write ͠ X γq = again. Multiplying both sides with the whole denominator of the right (note that this is surely a positive number) and arranging everything to the left side we get the following due to a lengthy computation: If ͠ p q = e , then every line which is not a 2-secant to  is contained completely in  (and the ideal point of it has weight two) since the affine points are single ones and an ideal point has weight at most two. Hence if there exists a double point in the ideal line then  has to be the union of two complete lines and otherwise every line is a 2-secant to  . In the first case we get a contradiction with Proposition 3.9 and in the latter case we get that the number of lines has to be equal to  ( ) | | 2 , but now  ͠ q | | = 2( + 1). Hence ͠ p q = e is not possible.
If ͠ p q < e , then the leading term in expression (7) is ͠ q p γ γ (−1 + 16 + 16 ) e 2 2 . If γ is chosen so that γ γ −1 + 16 + 16 < 0 2 and q and p e are large enough, then the leading term overflows the remaining ones, hence we will get a contradiction and conclude that our coloring must be trivial. The coefficient is negative if ≈ > 8 + 4 5 16, 944 γ 1 but then the remaining terms can be quite large. Thus at this point, we make a rather arbitrary choice of the parameters in our likes and, in case someone would need a differently set result, we will make a remark on the other possible choices.