Periodic Reporting for period 2 - LocalGlobal (Local vs Global Properties of Large Discrete Structures)
Okres sprawozdawczy: 2022-09-01 do 2024-02-29
Extremal combinatorics is one of the fastest growing areas of research within discrete mathematics.
Questions in this area deal with the asymptotic relations between various parameters of large discrete structures such
as graphs, hypegraphs, permutations, sets of integers, etc. This area has grown tremendously in the past few
decades, both in depth and in breadth, and supplied many spectacular results that affected various other areas of
mathematics, such as number theory, group theory, probability theory, information theory and theoretical com-
puter science. Many key insights that were developed in order to solve some of the core problems in extremal
combinatorics were later exported to other areas. Perhaps the prime example is Szemeredi’s theorem,
stating that dense sets of integers contain arbitrarily long arithmetic progressions. This theorem motivated some
of the most important investigations in extremal combinatorics such as the regularity method in graphs
and hypergraphs, the theory of quasi-random graphs and the theory of graph limits.
Szemeredi’s theorem also motivated the development of tools in other areas such as ergodic theory (the multiple recurrence theorem), harmonic analysis (the Gowers norms), number theory (the Green–Tao
theorem) and theoretical computer science (the PCP theorem and property testing). See also for a more detailed discussion.
To give the reader an idea of the type of questions we plan to investigate in this proposal, let us mention the
removal lemma of Ruzsa and Szemeredi, which is widely considered to be one of the cornerstone results
of extremal combinatorics. It states that an n-vertex graph with o(n^3) triangles can be made triangle free by
removing o(n^2) edges. The main motivation for proving this lemma was the following “innocent looking”
question, known as the (6,3)-problem: suppose we have a collection of subsets of a ground set V, such that
each e ∈E is of size 3 and the union of every triple of distinct subsets e1,e2,e3 ∈ contains at least 7 of the
points of V. How large can |E|be as a function of |V|? The surprising answer is that |E|= o(|V|^2), that is a
very simple local restriction (6 points do not form 3 sets) forces a very strong global one (E is relatively small).
Perhaps even more surprisingly, this lemma implies Szemeredi’s theorem for 3-term arithmetic progression
(i.e. Roth’s theorem). This lemma turned out to be extremely influential and motivated much of the
investigations mentioned in the previous paragraph. See for a comprehensive survey.
We propose to study various open problems in extremal combinatorics that relate local and global proper-
ties of various discrete structures. The first set of problems is directly related to the removal lemma mentioned
above. While a major goal is to improve the bounds for the hypergraph extension of this lemma, we also aim
to solve several related conjectures and open problems that have applications in theoretical computer science.
Another set of problems are motivated by questions in theoretical computer science from the area of property
testing. They ask to develop fast algorithms for deciding if a discrete structure satisfies some predetermined
property while observing only a small fraction of the input. We will develop a unifying theory for such problems using various tools introduced in the past few years. Along the way, we will expand the relation between
extremal combinatorics and theoretical computer science. Another set of problems asks to establish that
hypergraphs excluding certain types of sub-structures, possess surprisingly good Ramsey-type properties. Most of
these problems have been settled (at least partially) in the setting of graphs, and their hypergraph counterparts
are much more challenging.
The common thread going through many of the above problems (either implicitly or explicitly) is Szemeredi’s regularity lemma, which is one of the most powerful tools for studying problems in extremal
combinatorics. We aim to find new applications of the graph and hypergraph versions of this lemma, but more
importantly, to develop new efficient variants of the lemma, tailored for specific applications, that would thus
allow us to settle many of the problems discussed in this proposal.
Questions in this area deal with the asymptotic relations between various parameters of large discrete structures such
as graphs, hypegraphs, permutations, sets of integers, etc. This area has grown tremendously in the past few
decades, both in depth and in breadth, and supplied many spectacular results that affected various other areas of
mathematics, such as number theory, group theory, probability theory, information theory and theoretical com-
puter science. Many key insights that were developed in order to solve some of the core problems in extremal
combinatorics were later exported to other areas. Perhaps the prime example is Szemeredi’s theorem,
stating that dense sets of integers contain arbitrarily long arithmetic progressions. This theorem motivated some
of the most important investigations in extremal combinatorics such as the regularity method in graphs
and hypergraphs, the theory of quasi-random graphs and the theory of graph limits.
Szemeredi’s theorem also motivated the development of tools in other areas such as ergodic theory (the multiple recurrence theorem), harmonic analysis (the Gowers norms), number theory (the Green–Tao
theorem) and theoretical computer science (the PCP theorem and property testing). See also for a more detailed discussion.
To give the reader an idea of the type of questions we plan to investigate in this proposal, let us mention the
removal lemma of Ruzsa and Szemeredi, which is widely considered to be one of the cornerstone results
of extremal combinatorics. It states that an n-vertex graph with o(n^3) triangles can be made triangle free by
removing o(n^2) edges. The main motivation for proving this lemma was the following “innocent looking”
question, known as the (6,3)-problem: suppose we have a collection of subsets of a ground set V, such that
each e ∈E is of size 3 and the union of every triple of distinct subsets e1,e2,e3 ∈ contains at least 7 of the
points of V. How large can |E|be as a function of |V|? The surprising answer is that |E|= o(|V|^2), that is a
very simple local restriction (6 points do not form 3 sets) forces a very strong global one (E is relatively small).
Perhaps even more surprisingly, this lemma implies Szemeredi’s theorem for 3-term arithmetic progression
(i.e. Roth’s theorem). This lemma turned out to be extremely influential and motivated much of the
investigations mentioned in the previous paragraph. See for a comprehensive survey.
We propose to study various open problems in extremal combinatorics that relate local and global proper-
ties of various discrete structures. The first set of problems is directly related to the removal lemma mentioned
above. While a major goal is to improve the bounds for the hypergraph extension of this lemma, we also aim
to solve several related conjectures and open problems that have applications in theoretical computer science.
Another set of problems are motivated by questions in theoretical computer science from the area of property
testing. They ask to develop fast algorithms for deciding if a discrete structure satisfies some predetermined
property while observing only a small fraction of the input. We will develop a unifying theory for such problems using various tools introduced in the past few years. Along the way, we will expand the relation between
extremal combinatorics and theoretical computer science. Another set of problems asks to establish that
hypergraphs excluding certain types of sub-structures, possess surprisingly good Ramsey-type properties. Most of
these problems have been settled (at least partially) in the setting of graphs, and their hypergraph counterparts
are much more challenging.
The common thread going through many of the above problems (either implicitly or explicitly) is Szemeredi’s regularity lemma, which is one of the most powerful tools for studying problems in extremal
combinatorics. We aim to find new applications of the graph and hypergraph versions of this lemma, but more
importantly, to develop new efficient variants of the lemma, tailored for specific applications, that would thus
allow us to settle many of the problems discussed in this proposal.
Main results obtained.
1. We solved a 20 year old problem of Nathanson regrading certain extremal properties of the partition function.
2. We resolved a problem of Furedi and Ruszinko regarding a Turan problem on 3-uniform hypergraphs.
3. We resolved Problem 8 described in the project proposal.
4. We resolved Problem 9 described in the project proposal.
5. We resolved Problem 12 described in the project proposal.
6. We resolved Problem 15 described in the project proposal.
6. We resolved Problem 21 described in the project proposal.
1. We solved a 20 year old problem of Nathanson regrading certain extremal properties of the partition function.
2. We resolved a problem of Furedi and Ruszinko regarding a Turan problem on 3-uniform hypergraphs.
3. We resolved Problem 8 described in the project proposal.
4. We resolved Problem 9 described in the project proposal.
5. We resolved Problem 12 described in the project proposal.
6. We resolved Problem 15 described in the project proposal.
6. We resolved Problem 21 described in the project proposal.
We expect that by the end of the project we will resolve Problem 14.
We also expect to obtain new results related to the hypergraph regularity lemma.
We also expect to obtain new results related to the hypergraph regularity lemma.