Zarankiewicz Problem Research Articles

A new result on the problem of Zarankiewicz

Zarankiewicz ( Colloq. Math. 2 (1951) , 301) raised the following problem: Determine the least positive integer z ( m , n , j , k ) such that each 0–1-matrix with m rows and n columns containing z ( m , n , j , k ) ones has a submatrix with j rows and k columns consisting entirely of ones. This paper improves a result of Hylten-Cavallius ( Colloq. Math. 6 (1958) , 59–65) who proved: [ k 2 ] 1 2 ⩽ lim n→∞ inf z(n, n, 2, k)n − 3 2 ⩽ lim n→∞ sup z(n, n, 2, k)n − 3 2 ⩽ (k − 1) 1 2 . We prove that lim n→∞ z(n, n, 2, k)n − 3 2 exists and is equal to (k − 1) 1 2 . For the special case where k = 2 resp. k = 3 this result was proved earlier by Kövari, Sos and Turan ( Colloq. Math. 3 (1954) , 50–57) resp. Hylten-Cavallius.

A problem ? multi-colorings of graphs

In this paper we consider colorings of the edges of the complete graph Km with n colors such that the edges of any color form a non-trivial complete subgraph of Km. We allow an edge of Km to have more than one color. Such a coloring will be called r-admissible if no cycle of length r has a different color for each edge. Let E (m, n, r) be the maximum number of incidences of colors and edges, taken over all r-admissible colorings of Km with n colors. Then for r = 3,4, and 5 we give an upper bound for E (m, n, r); as well as a lower bound for E (m, n, r) for all r. An analogue to a problem of Zarankiewicz concerning 0, 1-matrices is mentioned.

A problem of Zarankiewicz

Zarankiewicz, in problem P 101, Colloq. Math., 2 (1951), p. 301, and others have posed the following problem: Determine the least positive integer k α, β ( m, n) so that if a 0,1-matrix of size m by n contains k α, β ( m, n) ones then it must have an α by β submatrix consisting entirely of ones. This paper improves upon previously known upper bounds for k α, β ( m, n) by proving that k αβ(m,n)⩾ 1+((β−1) ( p α−1 ))( m α ) + ((p+1)(α−1) α )n for each integer p greater than or equal to α − 1. Each of these inequalities is better than the others for a specific range of values of n. Equality is shown to hold infinitely often for each value of p. Finally some applications of this result are made to arrangements of lines in the projective plane.

The game of rectangles

A combinatorial problem is discussed, equivalent to a problem of Zarankiewicz, Upper and lower bounds are given for the number of pieces a player needs to win a certain board game.

On a problem of Knaster and Zarankiewicz

On a problem of Knaster and Zarankiewicz

Solution of the Zarankiewicz problem

Solution of the Zarankiewicz problem