On small -uniform hypergraphs with positive discrepancy
A two-coloring of the vertices of the hypergraph by red and blue has discrepancy if is the largest difference between the number of red and blue points in any edge. Let be the fewest number of edges in an -uniform hypergraph without a coloring with discrepancy . Erdős and Sós asked: is unbounded?
Keywords: hypergraph colorings, hypergraph discrepancy, prescribed matrix determinant.
A hypergraph is a pair , where is a finite set whose elements are called vertices and is a family of subsets of , called edges. A hypergraph is -uniform if every edge has size . A vertex -coloring of a hypergraph is a map .
The discrepancy of a coloring is the maximum over all edges of the difference between the number of vertices of two colors in the edge. The discrepancy of a hypergraph is the minimum discrepancy of a coloring of this hypergraph. Let be the minimal number of edges in an -uniform hypergraph (all edges have size ) having positive discrepancy.
Obviously, if then ; if but then . Erdős and Sős asked whether is bounded or not. N. Alon, D. J. Kleitman, C. Pomerance, M. Saks and P. Seymour  proved the following Theorem, showing in particular that is unbounded.
Let be an integer such that . Then
where stands for the least positive integer that does not divide .
To prove the upper bound they introduced several quantities. Let denote the set of all matrices with entries in such that the equation has exactly one non-negative solution (here stands for the vector with all entries equal to ). This unique solution is denoted . Let be the least integer such that is integer and let . For each positive integer , let be the least such that there exists a matrix with rows such that (obviously, because , where is the matrix with unit entries; is the identity matrix). The upper bound in (1) follows from the inequality for such that is odd.
Then N. Alon and V. H. Vũ  showed that for infinitely many . However they marked that trueness of inequality for arbitrary is not clear.
Our main result is the following
Let be a positive integer number. Then
for some constant .
Let be a positive integer number. Then
for some constant .
The main idea of the proof is to find a matrix with determinant and small entries satisfying some additional technical properties.
Let us consider the matrix and suppose that is not divisible on . Consider the system
It has the solution if and only if i. e. has prescribed residue modulo 19. Since is not divisible on , is not equal to zero modulo . So one can choose such that has prescribed residue modulo 19 and is odd.
Let us construct an -uniform hypergraph with positive discrepancy. Let be the solution of (3); note that , are positive and tend to with . Consider disjoint vertex sets of size and of size . If , then consider a vertex set of size disjoint with all sets , ; if let be a -vertex subset of . Let be a set if , otherwise . The edges of are listed:
Obviously, if has a coloring with discrepancy , then , where is the difference between blue and red verticed in , because the second edge can be reached by replacing on in the first edge. Similarly one can deduce that and for all pairs , . So one can put , . Because of the first edge we have . Obviously, and are odd numbers, so the minimal solution is , (or , which is the same because of red-blue symmetry). But then the last edge gives which contradicts with .
So we got an example if of an -uniform hypergraph with edges and positive discrepancy.
Proof of Theorem 1.2.
Let us denote by . We should construct a hypergraph with at most edges and positive discrepancy. Take such that . Then
Assume that is odd. Consider vectors in :
Note that the vector satisfies a system of linear equations
Choose odd such that is integer. Define
then the vector satisfies for , .
In the case we have and .
Choose so that and for . The solution is given by
Now let us construct a hypergraph in the following way: for let us take sets () of vertices of size such that all the sets are disjoint. Let the edge be the union of over and . By the choice of and we have . Then we add an edge
for every and for every such that . Clearly there are at most such edges. Finally, for every we add the edge
Summing up we have hypergraph with at most edges; at most of them have size not equal to . Let us correct this edges in the simplest way: if an edge has size less than then we add arbitrary vertices; if an edge has size greater than then we exclude arbitrary vertices.
Suppose that our hypergraph has discrepancy , so it has a proper coloring . For every set denote by the difference between the numbers of red and blue vertices of in . Obviously, because there are edges , such that . So we may write instead of .
If is odd then the vector satisfies
for some odd . It implies that
but this fails modulo . A contradiction. In the case we get a similar contradiction, as is not divisible by .
Thus we get a hypergraph on at most edges with positive discrepancy, the claim is proven. ∎
In fact, during the proof we have constructed a matrix of size of with bounded integer coefficients and with determinant . By Hadamard inequality, the determinant of matrix with bounded coefficients satisfies , thus , . We suppose that actually a matrix of size with bounded integer coefficients and determinant always exists; and moreover, it may be chosen satisfying additional properties which allow to replace the main estimate with (which asymptotically coincides with the lower bound).
It turns out, that for some values of (which depends only on ) a hypergraph, constructed from a matrix has the discrepancy separated from zero. In particular, in Example 2.1 the choice leads to the discrepancy 6.
Using Theorem 1.2 one can construct an -uniform hypergraph with discrepancy at least and edges (here stands for the nearest integer to ), as follows: let be a hypergraph realizing , be vertex-disjoint copies of . Let , . By the construction, every has discrepancy at least ; so by pigeonhole principle has discrepancy at least . Define . Finally, if , then exclude arbitrary vertices from every edge ; else add arbitrary vertices to every edge ; denote the result by . By definition , so the discrepancy of is at least . Since , we have
A. Raigorodskii independently asked the same question in a more general form: he introduced the quantity that is the minimal number of edges in a hypergraph without a vertex 2-coloring such that every edge has at least blue vertices and at least red vertices. So is the minimal number of a edges in a hypergraph with discrepancy at least , in particular .
Acknowledgements. The work was supported by the Russian Scientific Foundation grant 16-11-10014. The authors are grateful to A. Raigorodskii for the introduction to the problem, to N. Alon for directing our attention to the paper  and fruitful discussions and to N. Rastegaev for a very careful reading of the draft of the paper.
-  N. Alon, D. J. Kleitman, K. Pomerance, M. Saks, and P. Seymour. The smallest -uniform hypergraph with positive discrepancy. Combinatorica, 7(2):151–160, 1987.
-  Noga Alon and Văn H. Vũ. Anti-Hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs. Journal of Combinatorial Theory, Series A, 79(1):133–160, 1997.
-  D. D. Cherkashin and A. B. Kulikov. On two-colorings of hypergraphs. Doklady Mathematics, 83(1):68–71, 2011.
-  S. M. Teplyakov. Upper bound in the Erdős–Hajnal problem of hypergraph coloring. Mathematical Notes, 93(1-2):191–195, 2013.