Incidence Bounds for Block Designs
We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of incidences between points and -flats in affine geometries over finite fields. First, we show an upper bound on the number of incidences between sufficiently large subsets of the points and blocks of a BIBD. Second, we show that a sufficiently large subset of the points of a BIBD determines many -rich blocks. Third, we show that a sufficiently large subset of the blocks of a BIBD determines many -rich points. These last two results are new even in the special case of incidences between points and -flats in an affine geometry over a finite field.
As a corollary we obtain a tight bound on the number of -rich points determined by a set of points in a plane over a finite field, and use it to sharpen a result of Iosevich, Rudnev, and Zhai  on the number of triangles with distinct areas determined by a set of points in a plane over a finite field.
The structure of incidences between points and various geometric objects is of central importance in discrete geometry, and theorems that elucidate this structure have had applications to, for example, problems from discrete and computational geometry [18, 23], additive combinatorics [8, 13], harmonic analysis [17, 22], and computer science . The study of incidence theorems for finite geometry is an active area of research - e.g. [31, 7, 10, 20, 21, 28, 14].
The classical Szemerédi-Trotter theorem  bounds the maximum number of incidences between points and lines in -dimensional Euclidean space. Let be a set of points in and be a set of lines in . Let denote the number of incidences between points in and lines in . The Szemerédi-Trotter Theorem shows that . Ever since the original result, variations and generalizations of such incidence bounds have been intensively studied.
Incidence theorems for points and flats
If , then a result of Grosu  implies that we can embed and in without changing the underlying incidence structure. Then we apply the result from the complex plane, proved by Tóth  and Zahl , that . This matches the bound of Szemerédi and Trotter, and a well-known construction based on a grid of points in shows that the exponent of is tight.
The intermediate case of is rather poorly understood. A result of Bourgain, Katz, and Tao , later improved by Jones , shows that for . This result relies on methods from additive combinatorics, and is far from tight; in fact, we are not currently aware of any construction with that achieves .
For , we know tight bounds on . Using an argument based on spectral graph theory, Vinh  proved that , which gives both upper and lower bounds on . The upper bound meets the Szemerédi-Trotter bound of when , which is tight by the same construction used in the real plane. The lower bound becomes trivial for , and Vinh showed [?] that there are sets of points and lines such that there are no incidences between the points and lines. When , we have , which is tight when for a prime power (consider incidences between all points and lines in ).
In this paper we generalize Vinh’s argument to the purely combinatorial setting of balanced incomplete block designs (BIBDs). This shows that his argument depends only on the combinatorial structure induced by flats in the finite vector space. We apply methods from spectral graph theory. We both generalize known incidence bounds for points and flats in finite geometries, and prove results for BIBDs that are new even in the special case of points and flats in finite geometries. Finally, we apply one of these incidence bounds to improve a result of Iosevich, Rudnev, and Zhai  on the number of triangles with distinct areas determined by a set of points in .
In Section 2.1, we state definitions of and basic facts about BIBDs and finite geometries. In Section 2.2, we discuss our results on the incidence structure of designs. In Section 2.3, we discuss our result on distinct triangle areas in . In Section 3, we introduce the tools from spectral graph theory that are used to prove our incidence results. In Section 4, we prove the results stated in Section 2.2. In Section 5, we prove the result stated in Section 2.3.
1.2 Prior work
2.1 Definitions and Background
Let be a finite set (which we call the points), and let be a set of subsets of (which we call the blocks). We say that is an -BIBD if
each point is in blocks,
each block contains points,
each pair of points is contained in blocks, and
no single block contains all of the points.
It is easy to see that the following relations among the parameters of a BIBD hold:
The first of these follows from double counting the pairs such that . The second follows from fixing an element , and double counting the pairs such that .
In the case where is the set of all points in , and is the set of -flats in , we obtain a design with the following parameters :
The notation refers to the -binomial coefficient, defined for integers by
We will only use the fact that .
Given a design , we say that a point is incident to a block if . For subsets and , we define to be the number of incidences between and ; in other words,
Given a subset , we say that a point is -rich if it is contained in at least blocks of , and we define to be the number of -rich points in ; in other words,
Given a subset , we say that a block is -rich if it contains at least points of , and we define to be the number of -rich blocks in ; in other words,
2.2 Incidence Theorems
The first result on the incidence structure of designs is a generalization of the finite field analog to the Szemerédi-Trotter theorem proved by Vinh .
Let be an -BIBD. The number of incidences between and satisfies
Theorem 1 gives both upper and lower bounds on the number of incidences between arbitrary sets of points and blocks. The term corresponds to the number of incidences that we would expect to see between and if they were chosen uniformly at random. If is much larger than , then is much larger than . Thus the theorem says that every set of points and blocks determines approximately the “expected” number of incidences. When , the term on the right is larger, and Theorem 1 gives only an upper bound on the number of incidences. The Cauchy-Schwartz inequality combined with the fact that each pair of points is in at most blocks easily implies that . Hence, the upper bound in Theorem 1 is only interesting when .
In the case of incidences between points and -flats in , we get the following result as a special case of Theorem 1.
Let be a set of points and let be a set of -flats in . Then
Vinh  proved Corollary 2 in the case , and Bennett, Iosevich, and Pakianathan  derived the bounds for the remaining values of from Vinh’s bound, using elementary combinatorial arguments. Vinh’s proof is based on spectral graph theory, analogous to the proof of Theorem 1 that we present in Section 4. Cilleruelo proved a result similar to Vinh’s using Sidon sets .
We also show lower bounds on the number of -rich blocks determined by a set of points, and on the number of -rich points determined by a set of blocks.
While reading the statements of these theorems, it is helpful to recall from equation that .
Let be an -BIBD. Let and . Let with
Then, the number of -rich blocks is at least
Let be an -BIBD. Let and . Let , with
Then, the number of -rich points is at least
Theorem 3 is analogous to Beck’s theorem , which states that, if is a set of points in , then either points lie on a single line, or there are lines each contain at least points of , where and are fixed positive constants.
Both the case of Theorem 3 and Beck’s theorem are closely related to the de Bruijn-Erdős theorem [11, 26], which states that, if is a set, and is a set of subsets of such that each pair of elements in is contained in exactly members of , then either a single member of contains all elements of , or . Each of Theorem 3 and Beck’s theorem has an additional hypothesis on the de Bruijn-Erdős theorem, and a stronger conclusion. Beck’s theorem improves the de Bruijn-Erdős theorem when is a set of points in and is the set of lines that each contain points of . Theorem 3 improves the de Bruijn-Erdős theorem when is a sufficiently large subset of the points of a BIBD, and is the set of blocks that each contain at least points of .
As special cases of Theorems 3 and 4, we get the following results on the number of -rich points determined by a set of -flats in , and on the number of -rich -flats determined by a set of points in .
Let and . Let with
Then the number of -rich -flats is at least
Let and . Let be a subset of the -flats in with
Then, the number of -rich points is at least
For the case , the value of in Corollary 6 depends strongly on when . This dependence is necessary. For example, consider the case , i.e. lines in . Corollary 6 implies that a set of lines in determines -rich points, which is asymptotically fewer (with regard to ) than the total number of points in . This is tight, since the lines may lie in the union of two planes. By contrast, Corollary 5 implies that a set of points in determines -rich lines, which is a constant proportion of all lines in the space.
2.3 Distinct Triangle Areas
Iosevich, Rudnev, and Zhai  studied a problem on distinct triangle areas in . This is a finite field analog to a question that is well-studied in discrete geometry over the reals. Erdős, Purdy, and Strauss  conjectured that a set of points in the real plane determines at least distinct triangle areas. Pinchasi  proved that this is the case.
In , we define the area of a triangle in terms of the determinant of a matrix. Suppose a triangle has vertices , and , and let and denote the and coordinates of a point . Then, we define the area associated to the ordered triple to be the determinant of the following matrix:
Iosevich, Rudnev, and Zhai  showed that a set of at least points includes a point that is a common vertex of triangles having at least distinct areas. They first prove a finite field analog of Beck’s theorem, and then obtain their result on distinct triangle areas using this analog to Beck’s theorem along with some Fourier analytic and combinatorial techniques. Corollary 5 (in the case ) strengthens their analog to Beck’s theorem, and thus we are able to obtain the following strengthening of their result on distinct triangle areas.
Let . Let be a set of at least points in . Let be the set of triangles determined by . Then there is a point such that is a common vertex of triangles in with at least distinct areas, where is a positive constant depending only on , such that as .
Notice that Theorem 7 is tight in the sense that fewer than points might determine only triangles with area zero (if all points are collinear). It is a very interesting open question to determine the minimum number of points , such that any set of points of size determines all triangle areas. In fact, we are not currently aware of any set of more than points that does not determine all triangle areas.
3 Tools from Spectral Graph Theory
3.1 Context and Notation
Let be a -biregular bipartite graph; in other words, is a bipartite graph with left vertices , right vertices , and edge set , such that each left vertex has degree , and each right vertex has degree . Let be the adjacency matrix of , and let be the eigenvalues of . Let be the normalized second eigenvalue of .
Let be the number of edges in . For any two subsets of vertices and , denote by the number of edges between and . For a subset of vertices , denote by the set of vertices in that have at least neighbors in .
We will use two lemmas relating the normalized second eigenvalue of to its combinatorial properties. The first of these is the expander mixing lemma .
Lemma 8 (Expander Mixing Lemma).
Let with and let with . Then,
Several variants of this result appear in the literature, most frequently without the terms. For a proof that includes these terms, see , Lemma 4.15. Although the statement in  is not specialized for bipartite graphs, it is easy to modify it to obtain Lemma 8. For completeness, we include a proof in the appendix.
Let , and let . If such that
Let . Let , and let . We will calculate a lower bound on , from which we will immediately obtain a lower bound on .
Since each vertex in has at most edges to vertices in , we have . Along with the fact that , this gives
Lemma 8 implies that . Since we expect to be small, we will drop the term, and we have
By hypothesis, . Let such that . Then,
Define for . The derivative of is
Since , for any , we have . Hence, , and
Recall that , so this completes the proof. ∎
4 Proof of Incidence Bounds
Let be an -BIBD. Let be a bipartite graph with left vertices , right vertices , and if . Let be the adjacency matrix of . Then, the normalized second eigenvalue of is .
Let be the incidence matrix of ; that is, is a -valued matrix such that iff point is in block . We can write
Instead of analyzing the eigenvalues of directly, we’ll first consider the eigenvalues of . Since
is a block diagonal matrix, the eigenvalues of (counted with multiplicity) are the union of the eigenvalues of and the eigenvalues of . We will start by calculating the eigenvalues of .
The following observation about was noted by Bose .
where is the identity matrix and is the all-s matrix.
The entry corresponds to the number of blocks that contain both point and point . From the definition of an -BIBD, it follows that and if , and the conclusion of the proposition follows. ∎
We use the above decomposition to calculate the eigenvalues of .
The eigenvalues of are with multiplicity and with multiplicity .
The eigenvalues of are all . The eigenvalues of are with multiplicity and with multiplicity . The eigenvector of corresponding to eigenvalue is the all-ones vector, and the orthogonal eigenspace has eigenvalue . Since and share a basis of eigenvectors, the eigenvalues of are simply the sums of the corresponding eigenvalues of and . Hence, the largest eigenvalue of is , corresponding to the all-ones vector, and the remaining eigenvalues are , corresponding to vectors whose entries sum to . From equation , we have , and so we can write the largest eigenvalue as . ∎
Next, we use the existence of a singular value decomposition of to show that the nonzero eigenvalues of have the same values and occur with the same multiplicity as the eigenvalues of .
The following is a standard theorem from linear algebra; see e.g. [25, p. 429].
Theorem 13 (Singular value decomposition.).
Let be a real-valued matrix with rank . Then,
where is an orthogonal matrix, is an orthogonal matrix, and is a diagonal matrix. In addition, if the diagonal entries of are , then the nonzero eigenvalues of and are .
It is immediate from this theorem that the nonzero eigenvalues of , counted with multiplicity, are identical with those of . Hence, the nonzero eigenvalues of are with multiplicity and with multiplicity .
Clearly, the eigenvalues of are the squares of the eigenvalues of ; indeed, if is an eigenvector of with eigenvalue , then . Hence, the conclusion of the lemma will follow from the following proposition that the eigenvalues of are symmetric about . Although it is a well-known fact that the eigenvalues of the adjacency matrix of a bipartite graph are symmetric about , we include a simple proof here for completeness.
If is an eigenvalue of with multiplicity , then is an eigenvalue of with multiplicity .
Let and so that is an eigenvector of with corresponding nonzero eigenvalue .
Note that, since and and , we have that that and .
Hence, if is an eigenvalue of with eigenvector , then is an eigenvalue of with eigenvector . Since is a real symmetric matrix, it follows from the spectral theorem (e.g. [25, p. 227]) that has an orthogonal eigenvector basis; hence, we can match the eigenvectors of with eigenvalue with those having eigenvalue to show that the multiplicity of is equal to the multiplicity of . ∎
Now we can calculate that the nonzero eigenvalues of are and , each with multiplicity , and and , each with multiplicity . Hence, the normalized second eigenvalue of is , and the proof of Lemma 10 is complete. ∎
Proof of Theorem 1.
Proof of Theorem 3.
5 Application to Distinct Triangle Areas
Theorem 15 ().
Let . Suppose . Let, for ,
where . Then
We will also need the following consequence of Corollary 5.
Let and . There exists a constant , depending only on , such that the following holds.
Let be a set of points in . Then there is a point such that or more -rich lines are incident to . Moreover, if , then we can take .
By Corollary 5,
Denote by the number of incidences between points of and lines of . Since each line of is incident to at least points of , the average number of incidences with lines of that each point of participates in is at least
The derivative of with respect to is
Since this derivative is positive for and , we have that is a monotonically increasing function of for any fixed . Hence, for ,
For , let . Since the expected number of -rich lines incident to a point chosen uniformly at random is at least , there must be a point incident to so many -rich lines.
If , choose an arbitrary set of size . By the preceding argument, there must be a point incident to at least lines that are -rich in . Hence, is also incident to at least so many lines that are -rich in .
Proof of Theorem 7.
Let , so that . Let be as in Lemma 16. By Lemma 16, there is a point in incident to or more -rich lines. Let be a set of points such that there are exactly points of on exactly lines incident to . Clearly,
Let be translated so that is at the origin.
Each ordered pair corresponds to a triangle having as a vertex. By the definition of area, given in Section 2.3, the area of the triangle corresponding to is . For any point , let ; let . The area corresponding to is .
Hence, the number of distinct areas spanned by triangles with as a vertex is at least the number of distinct dot products . To write this in another way, let be as defined in Theorem 15 with and . Then, the number of distinct areas spanned by triangles containing is at least .
Since no line through the origin contains more than points of , Theorem 15 implies that
By Cauchy-Schwarz, the number of distinct triangle areas is at least
Hence, includes a point that is a vertex of triangles with at least
distinct areas. To complete the proof, check that has the claimed properties that for any , and that as . ∎
Appendix A Proof of Lemma 8
The proof here follows closely the proof of Lemma 4.15 in .
Let be the characteristic row vector of in ; in other words, is a vector of length with entries in such that iff vertex is in . Similarly, let be the characteristic vector of in . Note that