We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ with a fixed bipartition is an independent set with exactly $t$ vertices in each part; in other words, it is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, \Delta)$ be the largest $k$ for which every $n \times n$ bipartite graph with maximum degree $\Delta$ in one of the parts has a bi-hole of size $k$. Determining $f(n, \Delta)$ is thus the bipartite analogue of finding the largest independent set in graphs with a given number of vertices and bounded maximum degree. It has connections to the bipartite version of the Erdös--Hajnal conjecture, bipartite Ramsey numbers, and the Zarankiewicz problem. Our main result determines the asymptotic behavior of $f(n, \Delta)$. More precisely, we show that for large but fixed $\Delta$ and $n$ sufficiently large, $f(n, \Delta) = \Theta(\frac{\log \Delta}{\Delta} n)$. We further address more specific regimes of $\Delta$, especially when $\Delta$ is a small fixed constant. In particular, we determine $f(n, 2)$ exactly and obtain bounds for $f(n, 3)$, though determining the precise value of $f(n, 3)$ is still open.

predictive search (APS) is a new method by which systems can improve their search performance through experience. It is believed that the development of such methods is critical, as currently a tremendous number of computational results are potentially wasted by not integrating search and partial search results into the knowledge of a problem-solving system. In the APS model, pattern formation and associative recall are used to improve or replace search. In this paper, the theory, background, and motivations behind the model are presented and its application to two-player game-playing programs is discussed. In these programs, the system develops a knowledge base of patterns (boolean features) coupled with weights and, using pattern-oriented evaluation, performs only 1-ply search, yet competes respectably with programs that search more. The learning mechanism is a hybrid of machine learning and artificial intelligence techniques that have been successful in other settings. Specific examples and performance results are taken from the domains of Hexapawn, Tic-Tac-Toe, Pente, Othello, and chess.