## Final Report Summary - PROBCOMB (Sparse Discrete Structures)

The PI conducted research in extremal and probabilistic combinatorics. During the project he with several collaborators wrote 13 papers, 10 of them are still in the publication process. One of his collaborators (Hu) gave a lecture series (3 talks) on the use of flag algebras both at Szeged University and the Renyi Institute. This led to a collaboration with a local graduate student, Udvari, which paper is accepted for publication:

J. Balogh, P. Hu, B. Lidický, O. Pikhurko, B. Udvari, J. Volec Minimum number of monotone subsequences of length 4 in permutations, to appear in Combinatorics, Probability and Computing.

Additionally, using flag algebras, the PI with his collaborators solved an old problem of Erdos and Sos on edge-coloring of complete graphs

[ J. Balogh, P. Hu, B. Lidický, F. Pfender, J. Volec, M. Young, Rainbow triangles in three-colored graphs], and solved an other problem on the maximum induced densities of 5 -cycles [J. Balogh, P. Hu, B. Lidický, F. Pfender: Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle, submitted.]

The PI with collaborators investigated the structure and cardinality of combinatorial structures containing no copies of a given small structure. There are couple of important results achieved, here is the list of the most important ones: Paul Erdos suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on n vertices. The PI with Petrickova showed that the number of maximal triangle-free graphs is at most 2^{n^2/8+o(n^2)}, which matches the previously known lower bound.

[J. Balogh and S. Petrickova, The number of the maximal triangle-free graphs, Bull. London Math. Soc. (2014) 46 (5): 1003--1006.]

A related result is the following: Cameron and Erdos raised the question of how many maximal sum-free sets there are in {1, ... , n}, giving a lower bound of$2^{ n/4 }. The PI with Liu, Treglown and Sharifzadeh proved that there are in fact at most 2^{(1/4+o(1))n} maximal sum-free sets in that interval.

[ J. Balogh, H. Liu, M. Sharifzadeh, and A. Treglown, The number of maximal sum-free subsets of integers, to appear in Proceedings of AMS.

Using this general method, a problem of Osthus is solved as well: Let P(n) denote the power set of [n], ordered by inclusion, and let P (n,p) be obtained from P(n) by selecting elements from P (n) independently at random with probability p. A classical result of Sperner asserts that every antichain in P(n) has size at most that of the middle layer. The PI with Mycroft and Treglown proved an analogous result for P (n,p): If pn tends to infity then, with high probability, the size of the largest antichain in P(n,p) is at most (1+o(1)) p \binom{n}{ n/2 }$. Our condition on p is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of p.

[ J. Balogh, R. Mycroft and A. Treglown, A random version of Sperner's theorem, Journal of Combinatorial Theory, Series A, 128, (2014), 104--110].

Outreach activities:

The PI actively participated in high school mathematics education. He gave one central talk about his research at Szeged for all interested high school students who are attending regularly mathematical meetings. Additionally, whenever the PI did not travel, he was lecturing at a local high school (Sagvari Endre Gyakorlo Gimnazium) math club, which was held every Tuesday 13- 14, when the school was is season. There, the PI delivered a public talk at the Andras Hahn Memorial Ceremony.

At the Szeged University, the PI gave a special topics course on the Szemeredi Regularity Lemma.

With the researcher at charge (Peter Hajnal), he co-organized the combinatorics seminar, which had exceptionally high number of outsider visitors (about 15). In March 2014, the PI organized The Szeged University - Novi Sad University workshop in combinatorics which had 20+ talks, with about 40 people attending the talks. The PI hosted 4 graduate students from the University of Illinois (Liu, Hu, Delcourt, Sharifzadeh), each for 6 months, the PI got their visit funded by the University of Illinois.

J. Balogh, P. Hu, B. Lidický, O. Pikhurko, B. Udvari, J. Volec Minimum number of monotone subsequences of length 4 in permutations, to appear in Combinatorics, Probability and Computing.

Additionally, using flag algebras, the PI with his collaborators solved an old problem of Erdos and Sos on edge-coloring of complete graphs

[ J. Balogh, P. Hu, B. Lidický, F. Pfender, J. Volec, M. Young, Rainbow triangles in three-colored graphs], and solved an other problem on the maximum induced densities of 5 -cycles [J. Balogh, P. Hu, B. Lidický, F. Pfender: Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle, submitted.]

The PI with collaborators investigated the structure and cardinality of combinatorial structures containing no copies of a given small structure. There are couple of important results achieved, here is the list of the most important ones: Paul Erdos suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on n vertices. The PI with Petrickova showed that the number of maximal triangle-free graphs is at most 2^{n^2/8+o(n^2)}, which matches the previously known lower bound.

[J. Balogh and S. Petrickova, The number of the maximal triangle-free graphs, Bull. London Math. Soc. (2014) 46 (5): 1003--1006.]

A related result is the following: Cameron and Erdos raised the question of how many maximal sum-free sets there are in {1, ... , n}, giving a lower bound of$2^{ n/4 }. The PI with Liu, Treglown and Sharifzadeh proved that there are in fact at most 2^{(1/4+o(1))n} maximal sum-free sets in that interval.

[ J. Balogh, H. Liu, M. Sharifzadeh, and A. Treglown, The number of maximal sum-free subsets of integers, to appear in Proceedings of AMS.

Using this general method, a problem of Osthus is solved as well: Let P(n) denote the power set of [n], ordered by inclusion, and let P (n,p) be obtained from P(n) by selecting elements from P (n) independently at random with probability p. A classical result of Sperner asserts that every antichain in P(n) has size at most that of the middle layer. The PI with Mycroft and Treglown proved an analogous result for P (n,p): If pn tends to infity then, with high probability, the size of the largest antichain in P(n,p) is at most (1+o(1)) p \binom{n}{ n/2 }$. Our condition on p is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of p.

[ J. Balogh, R. Mycroft and A. Treglown, A random version of Sperner's theorem, Journal of Combinatorial Theory, Series A, 128, (2014), 104--110].

Outreach activities:

The PI actively participated in high school mathematics education. He gave one central talk about his research at Szeged for all interested high school students who are attending regularly mathematical meetings. Additionally, whenever the PI did not travel, he was lecturing at a local high school (Sagvari Endre Gyakorlo Gimnazium) math club, which was held every Tuesday 13- 14, when the school was is season. There, the PI delivered a public talk at the Andras Hahn Memorial Ceremony.

At the Szeged University, the PI gave a special topics course on the Szemeredi Regularity Lemma.

With the researcher at charge (Peter Hajnal), he co-organized the combinatorics seminar, which had exceptionally high number of outsider visitors (about 15). In March 2014, the PI organized The Szeged University - Novi Sad University workshop in combinatorics which had 20+ talks, with about 40 people attending the talks. The PI hosted 4 graduate students from the University of Illinois (Liu, Hu, Delcourt, Sharifzadeh), each for 6 months, the PI got their visit funded by the University of Illinois.