Final Report Summary - REGULARITY (Regularity and Irregularity in Combinatorics and Number Theory)
Graphs and hypergraphs are the most basic concepts in Combinatorics. They have application in a
wide range of Mathematics and Computer Science and also in other sciences. For example, the study of the large networks, actually a part of Combinatorics plays a major role in physics, biology, computer science, other branches of natural science and even in social sciences.. Some of the basic
notions in graph theory are the notion of a tree and a Hamiltonian path (or cycle). A graph is called a
tree if any two vertices are connected with exactly one path. This means with other words that a
graph is a tree if it is connected and it does not contain a cycle. A Hamiltonian path in a graph is a
path which visits every vertex exactly once. If it is a cycle too then we call it a Hamiltonian cycle. In
the project important new results were proved concerning the
Komlos-Loebl-Sos conjecture (actually a weaker form of the conjecture was justified) which
describes a condition under which a graph G would contain all trees of at most k edges as a subgraph.
In an other work a new and more effective proof for the Seymour's was found by a new method.
The conjecture gives a condition for a graph to contain a Hamiltonian cycle.
Another important method was found to solve the so called subset-sum problem, what is the maximal size of a subset of the first n integers so that its subset sum does not contain anarithmetic progression of length n.
Another result deals with the so called Heilbronn’s quadrangle problem and describes that given a suitable configuration of n points in the disc with radius 1, how large is at least the area of the convex hull of any 4 points. Of course it can be arbitrarily small but the point is to give a lower bound for the smallest area as a function of n.
Regularities and irregularities in the distribution of primes belong to the central problems of Number
Theory. While the most basic law for the distribution of primes (the Prime Number theorem
describing asymptotically the number of primes below a large number X) was found at the end of
nineteenth century, the more fine behavior of the distribution of primes, for example, many
problems dealing with the distribution of the differences between consecutive primes were
successfully studied only in the twentieth century culminating in the last 10 respectively 5 years
in very strong approximations of the famous twin prime problem. The twin prime conjecture asserts
that there are infinitely many prime pairs with a difference two. However even the weaker statement
that for any positive constant c there are consecutive prime pairs p and p' with a difference less than
c times the average difference logp was proved only few years ago by Goldston, Pintz and Yildirim.
It caused a sensation in the mathematical world when 5 years ago Y. Zhang, 6 months later by
another method J. Maynard and T. Tao proved that there are infinitely many consecutive prime pairs
with a difference less than 70 million (Zhang) and later 600 (Maynard). These results were further
improved finally to 246 by the so called Polymath8 project led by T. Tao with the interactive
participation of several mathematicians from many countries including two of our team members, G. Harcos and J. Pintz.
Later several 50-60-70 years old conjectures of Paul Erd#s were proved by these methods by J. Pintz
and others. To mention just one it was conjectured by Erdös 70 years ago that if we denote by d_n the nth difference between consecutive primes then the ratio d_n/d_(n+1) can take arbitrarily small and arbitrarily large values too. The methods of Zhang, Maynard and Tao are based on the earlier mentioned method of Goldston-Pintz-Yildirim (the so called GPY method).
wide range of Mathematics and Computer Science and also in other sciences. For example, the study of the large networks, actually a part of Combinatorics plays a major role in physics, biology, computer science, other branches of natural science and even in social sciences.. Some of the basic
notions in graph theory are the notion of a tree and a Hamiltonian path (or cycle). A graph is called a
tree if any two vertices are connected with exactly one path. This means with other words that a
graph is a tree if it is connected and it does not contain a cycle. A Hamiltonian path in a graph is a
path which visits every vertex exactly once. If it is a cycle too then we call it a Hamiltonian cycle. In
the project important new results were proved concerning the
Komlos-Loebl-Sos conjecture (actually a weaker form of the conjecture was justified) which
describes a condition under which a graph G would contain all trees of at most k edges as a subgraph.
In an other work a new and more effective proof for the Seymour's was found by a new method.
The conjecture gives a condition for a graph to contain a Hamiltonian cycle.
Another important method was found to solve the so called subset-sum problem, what is the maximal size of a subset of the first n integers so that its subset sum does not contain anarithmetic progression of length n.
Another result deals with the so called Heilbronn’s quadrangle problem and describes that given a suitable configuration of n points in the disc with radius 1, how large is at least the area of the convex hull of any 4 points. Of course it can be arbitrarily small but the point is to give a lower bound for the smallest area as a function of n.
Regularities and irregularities in the distribution of primes belong to the central problems of Number
Theory. While the most basic law for the distribution of primes (the Prime Number theorem
describing asymptotically the number of primes below a large number X) was found at the end of
nineteenth century, the more fine behavior of the distribution of primes, for example, many
problems dealing with the distribution of the differences between consecutive primes were
successfully studied only in the twentieth century culminating in the last 10 respectively 5 years
in very strong approximations of the famous twin prime problem. The twin prime conjecture asserts
that there are infinitely many prime pairs with a difference two. However even the weaker statement
that for any positive constant c there are consecutive prime pairs p and p' with a difference less than
c times the average difference logp was proved only few years ago by Goldston, Pintz and Yildirim.
It caused a sensation in the mathematical world when 5 years ago Y. Zhang, 6 months later by
another method J. Maynard and T. Tao proved that there are infinitely many consecutive prime pairs
with a difference less than 70 million (Zhang) and later 600 (Maynard). These results were further
improved finally to 246 by the so called Polymath8 project led by T. Tao with the interactive
participation of several mathematicians from many countries including two of our team members, G. Harcos and J. Pintz.
Later several 50-60-70 years old conjectures of Paul Erd#s were proved by these methods by J. Pintz
and others. To mention just one it was conjectured by Erdös 70 years ago that if we denote by d_n the nth difference between consecutive primes then the ratio d_n/d_(n+1) can take arbitrarily small and arbitrarily large values too. The methods of Zhang, Maynard and Tao are based on the earlier mentioned method of Goldston-Pintz-Yildirim (the so called GPY method).