Extremal K(s, t)-free bipartite graphs

Throughout this paper only undirected simple graphs (without loops or multiple edges) are considered. Unless stated otherwise, we follow the book by Bollobás [2] for undefined terminology and definitions. LetG = G(m,n) = G(X,Y ) denote a bipartite graph with vertex classesX and Y such that |X| = m and |Y | = n. A complete bipartite subgraph with s vertices in the class X and t vertices in the class Y is denoted byK(s,t). A bipartite graphG = G(X,Y ) containsK(s,t) as a subgraph if there exists an s-set S in X and a t-set T in Y such that the induced subgraph by S ∪ T in G, denoted G [S ∪ T ], is a complete bipartite K(s,t). Notice that a bipartite graph G may contain K(s,t) as a subgraph but it may be actually free ofK(t,s). A question that arises naturally is: what is the maximum number of edges, denoted by z(m,n; s, t), that a K(s,t)-free bipartite graph G = G(m,n) can have? Zarankiewicz [17] posed this problem for the particular case in which m = n and s = t = 3, shortly denoted by z(n; 3), when n = 4, 5, 6. This problem was solved by Sierpinski [15]. In the following years additional numerical values of the extremal function z(n; 3) for n ≥ 7 were provided by Brzezinski, see [16], Culik [4], Guy [11, 12, 13] and Guy and Znám [14]. Finally, the general problem of determining the exact value of the function z(m,n; s, t) as well as the family of extremal graphs has also been known as the Zarankiewicz problem. The extremal family for this problem, denoted by Z(m,n; s, t), is the set of bipartite graphs on m + n vertices with extremal size z(m,n; s, t), such that they do not containK(s,t) as a subgraph. †[email protected] ‡[email protected] §[email protected] ¶[email protected]