Summary: Discrete Mathematics: Methods and Challenges
Combinatorics is a fundamental mathematical discipline as well as an essential component of
many mathematical areas, and its study has experienced an impressive growth in recent years.
One of the main reasons for this growth is the tight connection between Discrete Mathematics and
Theoretical Computer Science, and the rapid development of the latter. While in the past many
of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the
modern theory has grown out of this early stage, and often relies on deep, well developed tools.
This is a survey of two of the main general techniques that played a crucial role in the development
of modern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated
by examples, focusing on the basic ideas and the connection to other areas.
The originators of the basic concepts of Discrete Mathematics, the mathematics of finite structures,
were the Hindus, who knew the formulas for the number of permutations of a set of n elements, and
for the number of subsets of cardinality k in a set of n elements, already in the sixth century. The
beginning of Combinatorics as we know it today started with the work of Pascal and De Moivre in
the 17th century, and continued in the 18th century with the seminal ideas of Euler in Graph Theory,
with his work on partitions and their enumeration, and with his interest in latin squares. These old
results are among the roots of the study of formal methods of enumeration, the development of