The Zarankiewicz Problem via Chow Forms
نویسندگان
چکیده
The well-known Zarankiewicz problem [Za] is to determine the least positive integer Z(m,n, r, s) such that each m × n 0-1 matrix containing Z(m,n, r, s) ones has an r × s submatrix consisting entirely of ones. In graph-theoretic language, this is equivalent to finding the least positive integer Z(m, n, r, s) such that each bipartite graph on m black vertices and n white vertices with Z(m,n, r, s) edges has a complete bipartite subgraph on r black vertices and s white vertices. A complete solution of the Zarankiewicz problem has not been given. While exact values of Z(m,n, r, s) are known for certain infinite subsets of m,n, r and s, only asymptotic bounds are known in the general case; for example, see Čuĺık [Č], Füredi [F], Guy [G], Hartmann, Mycielski and Ryll-Nardzewski [HMR], Hyltén-Cavallius [HC], Irving [I], Kövári, Sós and Turán [KST], Mörs [M], Reiman [Re], Roman [Ro], Znám [Zn]. Even the case r = s = 2 has not been answered in general. Here we quote some known facts about this case: Hartmann, Mycielski and Ryll-Nardzewski [HMR] proved the asymptotic bounds
منابع مشابه
Spectral Extrema for Graphs: The Zarankiewicz Problem
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 (
متن کاملExtremal Graph Theory
3 Third Lecture 11 3.1 Applications of the Zarankiewicz Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Turán Problem for Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 The Girth Problem and Moore’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Application of Moore’s Bound to Graph Spanners . . . . . . . . . . . ....
متن کاملCutting 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 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...
متن کاملSeparating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem
For every positive integer k, we construct an explicit family of functions f : {0, 1}n → {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 Ω ( n k42k ) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our const...
متن کاملOn the half?half case of the Zarankiewicz problem
Consider the minimum number f(m,n) of zeroes in a 2m×2n (0, 1)-matrixM 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 limm→∞{f(m,m+ 1)/m}. In this paper, we sh...
متن کامل