Extremal combinatorics and asymptotic enumeration

Combinatorics is a branch of mathematics studying finite structures. The generality of these questions suggests wide applicability of combinatorics in other areas of pure mathematics (most notably in algebra, number theory, probability, and topology), as well as in real-world applications (discrete optimization, computer science).One of the oldest and most central parts of combinatorics are graph theory and enumerative combinatorics. Graph theory models networks (such as road connections, or internet users), and enumerative combinatorics concerns studying counting questions of various kinds.Extremal graph theory is a broad part of graph theory which investigates interplay between various graph parameters. One of the main tools in Extremal graph theory is the so-called Szemeredi Regularity Lemma. This tool (developed in the 70's) has become one of the corner-stones of modern mathematics. Recently, using the insights gained from the Regularity Lemma, Lovasz and Szegedy initiated study of graph limits.The proposed research project addresses major open questions in extremal graph theory and aims contribute to general theories the Regularity Lemma, graph limits, and by developing novel tools which will be used in enumerative combinatorics.

The proposed research project is of theoretical nature. The primary beneficiary will be the European and especially the British community in Discrete Mathematics. It has been pointed out in recent International Review of UK Research in Mathematics (a review undertaken on behalf of the EPSRC and CMS, 2004), that even though there is a significant growth in research on discrete mathematics and its applications to computer science, this area is still under-represented in the UK, both in Mathematics and Computer Science Departments . The research in this proposal can contribute to this undertaking. On the other hand, it should be emphasized that combinatorics, and graph theory in particular are widely applicable in real-world problems, and techniques developed to address theoretical problems have found applications in industry. Let us name applications in design of or operations with networks of various kind (which employ a variety of graph-theoretic results), or applications in coding theory (where the so-called expander graphs are used - objects very close to my main research interests), a field which for example designs optimal ways how to store information on CDs. The knowledge gained during my studies in Computer Science will help me link my results to Theoretical Computer Science, in particular to property testing and analysis of randomized algorithms. Activities at the Centre for Discrete Mathematics and Its Applications at the University of Warwick will allow me to bring my theoretical research in industrial practice; this is one of the main of missions of the Centre. The applications of my mathematical results in Computer Science will be made available in a form accessible to the CS community. In particular, I will publish these results in renowned Computer Science conferences, for example in ACM-SIAM Symposium on Discrete Algorithms (SODA), Annual IEEE Symposium on Foundations of Computer Science (FOCS), or ACM Symposium on Theory of Computing (STOC).