Zarankiewicz Problem Research Articles

On the Zarankiewicz Problem for Graphs with Bounded VC-Dimension

On the Zarankiewicz Problem for Graphs with Bounded VC-Dimension

Ramsey numbers and the Zarankiewicz problem

AbstractBuilding on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form for a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an by ‐matrix that does not have any matrix from a fixed finite family derived from as a submatrix. As an application, we give new lower bounds for the Ramsey numbers and , namely, and . We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of for any fixed integer .

Some remarks on the Zarankiewicz problem

AbstractThe Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.

Cutting lemma and Zarankiewicz’s problem in distal structures

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in o-minimal expansions of fields. Using it, we generalize the results in Fox et al. (J Eur Math Soc 19(6):1785–1810, 2017 ) on the semialgebraic planar Zarankiewicz problem to arbitrary o-minimal structures, in particular obtaining an o-minimal generalization of the Szemeredi–Trotter theorem.

The Zarankiewicz problem in 3‐partite graphs

AbstractLet be a graph, be an integer, and write for the maximum number of edges in an ‐vertex graph that is ‐partite and has no subgraph isomorphic to . The function has been studied by many researchers. Finding is a special case of the Zarankiewicz problem. We prove an analog of the Kövári‐Sós‐Turán theorem for 3‐partite graphs by showing urn:x-wiley:10638539:media:jcd21654:jcd21654-math-0009 for . Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound is asymptotically best possible in the case that and is odd, that is, for . In the cases of and , we use a result of Allen, Keevash, Sudakov, and Verstraëte, to show that a similar upper bound holds for all and gives a better constant when . Finally, we point out an interesting connection between difference families from design theory and .

Zarankiewicz's problem for semi-algebraic hypergraphs

Zarankiewicz's problem asks for the largest possible number of edges in a graph that does not contain a Ku,u subgraph for a fixed positive integer u. Recently, Fox, Pach, Sheffer, Sulk and Zahl [12] considered this problem for semi-algebraic graphs, where vertices are points in Rd and edges are defined by some semi-algebraic relations. In this paper, we extend this idea to semi-algebraic hypergraphs. For each k≥2, we find an upper bound on the number of hyperedges in a k-uniform k-partite semi-algebraic hypergraph without Ku1,…,uk for fixed positive integers u1,…,uk. When k=2, this bound matches the one of Fox et al. and when k=3, it isO((mnp)2d2d+1+ε+m(np)dd+1+ε+n(mp)dd+1+ε+p(mn)dd+1+ε+mn+np+pm), where m,n,p are the sizes of the parts of the tripartite hypergraph and ε is an arbitrarily small positive constant. We then present applications of this result to a variant of the unit area problem, the unit minor problem and intersection hypergraphs.

A semi-algebraic version of Zarankiewicz's problem

A bipartite graph G is semi-algebraic in \mathbb{R}^d if its vertices are represented by point sets P,Q \subset \mathbb{R}^d and its edges are defined as pairs of points (p,q) \in P \times Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k , the maximum number of edges in a K_{k,k} -free semi-algebraic bipartite graph G = (P,Q,E) in \mathbb{R}^2 with |P| = m and |Q| = n is at most O((mn)^{2/3} + m + n) , and this bound is tight. In dimensions d \geq 3 , we show that all such semi-algebraic graphs have at most C\left((mn)^{ \frac{d}{d+1} + \epsilon} + m + n\right) edges, where here \epsilon is an arbitrarily small constant and C = C(d,k,t,\epsilon) . This result is a far-reaching generalization of the classical Szemerédi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials. We also present various applications of our theorem. For example, a general point-variety incidence bound in \mathbb R^d , an improved bound for a d -dimensional variant of the Erdös unit distances problem, and more.

On the Zarankiewicz problem for intersection hypergraphs

Let d and t be fixed positive integers, and let Kt,…,td denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős [6], the number of hyperedges of a d-uniform hypergraph on n vertices that does not contain Kt,…,td as a subhypergraph, is nd−1td−1. This bound is not far from being optimal.We address the same problem restricted to intersection hypergraphs of (d−1)-dimensional simplices in Rd. Given an n-element set S of such simplices, let Hd(S) denote the d-uniform hypergraph whose vertices are the elements of S, and a d-tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if Hd(S) does not contain Kt,…,td as a subhypergraph, then its number of edges is O(n) if d=2, and O(nd−1+ϵ) for any ϵ>0 if d≥3. This is almost a factor of n better than Erdős's above bound. Our result is tight, apart from the error term ϵ in the exponent. In particular, for d=2, we obtain a theorem of Fox and Pach [7], which states that every Kt,t-free intersection graph of n segments in the plane has O(n) edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses size-sensitive cuttings, a variant of random sampling.

Maxima of the Q-index: Graphs with no Ks,t

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order n that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases.More precisely, it is shown that if G is a graph of order n, with no subgraph isomorphic to K2,s+1, then the largest eigenvalue q(G) of the signless Laplacian of G satisfiesq(G)≤n+2s2+12(n−2s)2+8s, with equality holding if and only if G is a join of K1 and an s-regular graph of order n−1.

On the Edit Distance from K2,t-Free Graphs

The edit distance between two graphs on the same vertex set is defined to be the size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, H, is a function of p, and measures, asymptotically, the furthest graph of edge density p from H under this metric. In this article, we address the hereditary property Forb K2,t, the property of having no induced copy of the complete bipartite graph with two vertices in one class and t in the other. Employing an assortment of techniques and colored regularity graph constructions, we are able to determine the edit distance function over the entire domain p∈[0,1] when t=3,4 and extend the interval over which the edit distance function for Forb K2,t is known for all values of t, determining its maximum value for all odd t. We also prove that the function for odd t has a nontrivial interval on which it achieves its maximum. These are the only known principal hereditary properties for which this occurs. In the process of studying this class of functions, we encounter some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz.

Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

The de Bruijn–Erdős theorem for hypergraphs

Fix integers n ≥ r ≥ 2. A clique partition of \({{[n] \choose r}}\) is a collection of proper subsets \({A_1, A_2, \ldots, A_t \subset [n]}\) such that \({\bigcup_i{A_i \choose r}}\) is a partition of \({{[n]\choose r}}\) . Let cp(n, r) denote the minimum size of a clique partition of \({{[n] \choose r}}\) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3,$${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$We conjecture cp(n, r) = (1 + o(1))n r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1 + o(1))n r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n r) of the r-sets of [n] are covered. We also give an absolute lower bound \({{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}\) when n = q 2 + q + r − 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

A contribution to the Zarankiewicz problem

Given positive integers m , n , s , t , let z ( m , n , s , t ) be the maximum number of ones in a ( 0 , 1 ) matrix of size m × n that does not contain an all ones submatrix of size s × t . We show that if s ⩾ 2 and t ⩾ 2 , then for every k = 0 , … , s - 2 , z ( m , n , s , t ) ⩽ ( s - k - 1 ) 1 / t nm 1 - 1 / t + kn + ( t - 1 ) m 1 + k / t . This generic bound implies the known bounds of Kövari, Sós and Turán, and of Füredi. As a consequence, we also obtain the following results: Let G be a graph of n vertices and e ( G ) edges, and let μ be the spectral radius of its adjacency matrix. If G does not contain a complete bipartite subgraph K s , t , then the following bounds hold μ ⩽ ( s - t + 1 ) 1 / t n 1 - 1 / t + ( t - 1 ) n 1 - 2 / t + t - 2 , and e ( G ) < 1 2 ( s - t + 1 ) 1 / t n 2 - 1 / t + 1 2 ( t - 1 ) n 2 - 2 / t + 1 2 ( t - 2 ) n .

Spectral Extrema for Graphs: The Zarankiewicz Problem

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

New results on the Zarankiewicz problem

Let ( X , Y ) denote a bipartite graph with classes X and Y such that | X | = m and | Y | = n . A complete bipartite subgraph with s vertices in X and t vertices in Y is denoted by K ( s , t ) . The Zarankiewicz problem consists in finding the maximum number of edges, denoted by z ( m , n ; s , t ) , of a bipartite graph ( X , Y ) with | X | = m and | Y | = n without a complete bipartite K ( s , t ) as a subgraph. First, we prove that z ( m , n ; s , t ) = mn - ( m + n - s - t + 1 ) if max { m , n } ⩽ s + t - 1 . Then we characterize the family Z ( m , n ; s , t ) of extremal graphs for the values of parameters described above. Finally, we study the s = t case. We give the exact value of z ( m , n ; t , t ) if 2 t ⩽ n ⩽ 3 t - 1 and we characterize the extremal graphs if either n = 2 t or both 2 t < n ⩽ 3 t - 1 and m ⩽ ⌊ ( 3 t - 1 ) / 2 ⌋ .

Extremal K(s,t)-free bipartite graphs

Abstract We approach a well-known topic in extremal graph theory, the so-called Zarankiewicz Problem [K. Zarankiewicz, Problem P 101, Colloq. Math. 2 (1951), 301], which consists in: (i) calculating the maximum number of edges z ( m , n ; s , t ) that a bipartite graph G with partite classes of cardinalities m and n can have such that G is free of a complete bipartite subgraph K ( s , t ) with s vertices in the m class and t vertices in the n class; (ii) describing all the corresponding extremal bipartite graphs having that maximum number of edges. In this paper, the exact value of z ( m , n ; s , t ) is calculated and the corresponding family Z ( m , n ; s , t ) of extremal graphs is characterized when the parameters satisfy certain relationships.

On the half–half case of the Zarankiewicz problem

Consider the minimum number f(m,n) zeroes in a 2m×2n (0,1) -matrix M that contains no m×n submatrix of ones. This special case of the well-known Zarankiewicz problem was studied by Griggs and Ouyang, who showed, for m⩽n, that 2n+m+1⩽f(m,n)⩽2n+2m− gcd(m,n)+1 . The lower bound is sharp when m is fixed for all large n. They proposed determining lim m→∞{f(m,m+1)/m} . In this paper, we show that this limit is 3. Indeed, we determine the actual value of f(m,km+1) for all k,m. For general m,n, we derive a new upper bound on f(m,n). We also give the actual value of f(m,n) for all m⩽7 and n⩽20.

Contributions to the problem of Zarankiewicz

Let G 2 ( n ) denote a bipartite graph with n vertices in each color class, and let z ( n , t ) be the bipartite Turán number, representing the maximum possible number of edges in G 2 ( n ) if it is not to contain a copy of the complete bipartite subgraph K ( t , t ). It is then clear that ζ ( n , t )= n 2 − z ( n , t ) denotes the smallest possible number of zeros in an n × n zero–one matrix that does not contain a t × t submatrix consisting of all ones. We investigate the behaviour of z ( n , t ) when n goes to infinity (and perhaps t →∞ as well). The case t= o (n 1/3 );t⪢ log n has been considered by Godbole et al. [Electron. J. Combin. 4 (1997) 14pp], and we focus here on the overlapping case 2⩽ t ⪡ n 1/5 . Fill an n × n matrix randomly with z ones and n 2 − z = ζ zeros. Then, we prove that if t →∞, the asymptotic probability that there are no t × t submatrices with all ones is one or zero, according as z=n 2−2/t t e 2/t {1+ o (1)} or z=n 2−2/t t e 2/t {1+ o ∗ (1)}, where the o(1) and o ∗ (1) functions may be taken to be ( log t−a n )/t 2 and ( log t+a n )/t 2 respectively, for any a n that tends to infinity. If t is finite, a less sharp result is shown to be valid. The proof employs the Janson exponential inequalities. Finally, the Stein–Chen method is employed to derive a Poisson approximation for the probability distribution of the number of copies of K ( t , t ) in the random bipartite graph; an extreme value limit for the probability that there are no t × t submatrices with all ones, when the value of z is at the threshold level, is derived as a corollary.

A very simple function that requires exponential size read-once branching programs

A Boolean function in n variables is presented that is computable in depth 2 monotone AC 0 and has prime implicants of length 2 only but requires 2 Ω(√n) size read-once branching programs. The function considered is defined using (1, 1)-disjoint Boolean sums and the solution of the famous problem of Zarankiewicz.

(0,1)-Matrices with No Half–Half Submatrix of Ones

We consider the minimum number of zeroes in a 2m× 2n(0,1)-matrixMthat contains nom×nsubmatrix of ones. We show that this number, denoted byf(m,n), is at least 2n+m+ 1 form≤n. We determine exactly when this bound is sharp and determine the extremal matrices in these cases. For anym, the bound is sharp forn=mand for all but finitely manyn>m. A general upper bound due to Gentry,f(m,n) ≤ 2m+ 2n−gcd(m,n) + 1, is also derived. Our problem is a special case of the well-known Zarankiewicz problem.