the zarankiewicz number z(b; s) is the maximum size of a subgraph of kb,b which does not contain ks,s as a subgraph. the two-color bipartite ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of kb,b with two colors contains a ks,s in the rst color or a kt,t in the second color.in this work, we design and exploit a computational method for bounding and computin...

Abstract: The well known Zarankiewicz’ conjecture is said that the crossing number of the complete bipartite graph Km,n (m ≤ n) is Z(m, n), where Z(m,n) = ⌊ m 2 ⌋⌊ 2 ⌋⌊ 2 ⌋ ⌊ 2 ⌋ (for any real number x, ⌊x⌋ denotes the maximal integer no more than x). Presently, Zarankiewicz’ conjecture is proved true only for the case m ≤ 6. In this article, the authors prove that if Zarankiewicz’ conjecture h...

The Zarankiewicz number z(s, m) is the maximum number of edges in a subgraph of K(s, s) that does not contain K(m, m) as a subgraph. The bipartite Ramsey number b(m, n) is the least positive integer b such that if the edges of K(b, b) are coloured with red and blue, then there always exists a blue K(m, m) or a red K(n, n). In this paper we calculate small exact values of z(s, 2) and determine b...

Let d and t be fixed positive integers, and let K t,...,t 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 [7], the number of hyperedges of a d-uniform hypergraph on n vertices that does not co...

The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...

The Zarankiewicz number z(m,n; s, t) is the maximum number of edges in a subgraph of Km,n that does not contain Ks,t as a subgraph. The bipartite Ramsey number b(n1, · · · , nk) is the least positive integer b such that any coloring of the edges of Kb,b with k colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1 ≤ i ≤ k. If ni = m for all i, then we denote this ...

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 ominimal expansions of fields. Using it, we generalize the results in [10] on the semialgebraic planar Zarankiewicz problem to arbitrary o-minimal structures, in particular obtaining an o-minimal generalizatio...

Because of the large success of very large scale integration (VLSI) technology many researchers have focused on optimizing the VLSI circuit layout. One of the major tasks is minimizing the number of wire crossings in a circuit, as this greatly reduces the chance of cross-talk in long crossing wires carrying the same signal and also allows for faster operation and less power dissipation. The que...

Let G be a graph on n vertices with spectral radius λ (this is the largest eigenvalue of the adjacency matrix of G). We show that if G does not contain the complete bipartite graph Kt,s as a subgraph, where 2 6 t 6 s, then λ 6 (

The crossing number of a graph G is the minimum number of crossings of its edges among the drawings of G in the plane and is denoted by cr(G). Zarankiewicz conjectured that the crossing number of the complete bipartite graph Km,n equals . This conjecture has been verified by Kleitman for min {m, n} ≤ 6. Using this result, we give the exact values of crossing number of the join of a certain grap...