Van der Waerden's theorem states that any coloring of with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number which is the smallest such that any -coloring of guarantees the presence of a monochromatic arithmetic progression of length …
For a graph , the Turán number is the maximum number of edges in an -vertex simple graph not containing . The celebrated Erdős-Stone-Simonovits Theorem gives that \ where is the chromatic number of . This theorem asymptotically solves the problem when . In case of bipartite graphs , not even the order of magnitude …
We determine the maximum number of copies of in a -free -vertex graph for all integers and sufficiently large . Moreover, for and any integer we obtain the maximum number of cycles of length in an -vertex -free bipartite graph. This is joint work with Ervin Győri (Renyi …
In computer science, random expressions are commonly used to analyze algorithms, either to study their average complexity, or to generate benchmarks to test them experimentally. In general, these approaches only consider the expressions as purely syntactic trees, and completely ignore their semantics — i.e. the mathematical object represented by the expression. However, two different expressions …