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It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight., Appendices 1-5 are contained in the ancillary pdf file available on arXiv for download

Komlos conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.

Alon, Balogh, Keevash, and Sudakov proved that the $(k-1)$-partite Turan graph maximizes the number of distinct $r$-edge-colorings with no monochromatic $K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this paper, we determine this function asymptotically for $r=2$ among $n$-vertex graphs with a sublinear independence number. Somewhat surprisingly, unlike Alon, Balog, Keevash, and Sudakov's result, the extremal construction from Ramsey--Turan theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with a sublinear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an $n$-vertex $K_k$-free graph $G$ with $\alpha(G)=o(n)$. The extremal graphs have a similar structure to the extremal graphs for the classical Ramsey--Turan problem, i.e., when the number of edges is maximized.

For a sequence $(H_i)_{i=1}^k$ of graphs, let $\textrm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$. When each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $\textrm{nim}(n;H_1,\ldots, H_k)/{n\choose 2}$ as $n\to\infty$ and prove the corresponding stability result. Furthermore, if each $H_i$ is what we call \emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $\textrm{nim}(n;H_1,\ldots, H_k)$ exactly for all sufficiently large~$n$. The special case $\textrm{nim}(n;K_3,K_3,K_3)$ of our result answers a question of Ma. For bipartite graphs, we mainly concentrate on the two-colour symmetric case (i.e., when $k=2$ and $H_1=H_2$). It is trivial to see that $\textrm{nim}(n;H,H)$ is at least $\textrm{ex}(n,H)$, the maximum size of an $H$-free graph on $n$ vertices. Keevash and Sudakov showed that equality holds if $H$ is the $4$-cycle and $n$ is large; recently Ma extended their result to an infinite family of bipartite graphs. We provide a larger family of bipartite graphs for which $\textrm{nim}(n;H,H)=\textrm{ex}(n,H)$. For a general bipartite graph $H$, we show that $\textrm{nim}(n;H,H)$ is always within a constant additive error from $\textrm{ex}(n,H)$, i.e.,~$\textrm{nim}(n;H,H)= \textrm{ex}(n,H)+O_H(1)$., 24 pages, 3 figures

We show that, in contrast to the integers setting, almost all even order abelian groups $G$ have exponentially fewer maximal sum-free sets than $2^{\mu(G)/2}$, where $\mu(G)$ denotes the size of a largest sum-free set in $G$. This confirms a conjecture of Balogh, Liu, Sharifzadeh and Treglown., Comment: 14 pages

The classical Corr adi-Hajnal theorem claims that every n-vertex graph G with (G) 2n=3 contains a triangle factor, when 3jn. In this paper we asymptotically determine the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n-vertex graph with (G) = o(n) and (G) (1=2 + o(1))n, then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratic many edges, and (G) (1=2 +o(1))n, then G has a Kr-factor for n suciently large. We also propose many related open problems whose solutions could show a relationship with RamseyTur an theory. Additionally, we also consider a fractional variant of the Corr adi-Hajnal Theorem, settling a conjecture of Balogh-Kemkes-Lee-Young. Let t2 (0; 1) and w : E(Kn)! [0; 1]. We call a triangle in Kn heavy if the sum of the weights on its edges is more than 3t. We prove that if 3jn and w is such that for every vertex v the sum of w(e)

In 1987, Kolaitis, Promel and Rothschild proved that, for every fixed r∈N, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r ≤ (logn)1/4. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobas and Simonovits.

Cameron and Erdős [P. Cameron and P. Erdős, Notes on sum-free and related sets, Combin. Probab.Comput., 8, (1999), 95–107] asked whether the number of maximal sum-free sets in { 1 , … , n } is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2 ⌊ n / 4 ⌋ for the number of maximal sum-free sets. We prove the following: For each 1 ≤ i ≤ 4 , there is a constant C i such that, given any n ≡ i mod 4 , { 1 , … , n } contains ( C i + o ( 1 ) ) 2 n / 4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [B. Green, The Cameron-Erdős conjecture, Bull. London Math. Soc., 36, (2004), 769–778; B. Green, A Szemeredi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal., 15, (2005), 340–376], a structural result of Deshouillers, Freiman, Sos and Temkin [J. Deshouillers, G. Freiman, V. Sos and M. Temkin, On the structure of sumfree sets II, Asterisque, 258, (1999), 149–161] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [B. Green, R. Morris, Counting sets with small sumset and applications, Combinatorica, to appear].

Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely many values of $n$, the number of subsets of $\{1,2,\ldots, n\}$ that do not contain a $k$-term arithmetic progression is at most $2^{O(r_k(n))}$, where $r_k(n)$ is the maximum cardinality of a subset of $\{1,2,\ldots, n\}$ without a $k$-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of $n$, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on $r_k(n)$ to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size $\Theta(r_k(n))$ contain superlinearly many $k$-term arithmetic progressions. For integers $r$ and $k$, Erd\Ho s asked whether there is a set of integers $S$ with no $(k+1)$-term arithmetic progression, but such that any $r$-coloring of $S$ yields a monochromatic $k$-term arithmetic progression. Ne\v{s}et\v{r}il and R\"odl, and independently Spencer, answered this question affirmatively. We show the following density version: for every $k\ge 3$ and $\delta>0$, there exists a reasonably dense subset of primes $S$ with no $(k+1)$-term arithmetic progression, yet every $U\subseteq S$ of size $|U|\ge\delta|S|$ contains a $k$-term arithmetic progression. Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea., Comment: To appear in International Mathematics Research Notices. This is a longer version than the journal version, containing two additional minor applications of the container method

Komlos conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph K d + 1 minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when d is sufficiently large. In fact we prove a stronger result: for large d, any graph G with average degree at least d contains almost twice as many Hamiltonian subsets as K d + 1 , unless G is isomorphic to K d + 1 or a certain other graph which we specify.

Given a graph $F$, let $s_t(F)$ be the number of subdivisions of $F$, each with a different vertex set, which one can guarantee in a graph $G$ in which every edge lies in at least $t$ copies of $F$. In 1990, Tuza asked for which graphs $F$ and large $t$, one has that $s_t(F)$ is exponential in a power of $t$. We show that, somewhat surprisingly, the only such $F$ are complete graphs, and for every $F$ which is not complete, $s_t(F)$ is polynomial in $t$. Further, for a natural strengthening of the local condition above, we also characterise those $F$ for which $s_t(F)$ is exponential in a power of $t$., Comment: 7 pages, to appear in Combinatorics, Probability and Computing

Recently, settling a question of Erd\H{o}s, Balogh and Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest., Comment: 17 pages

Mader conjectured that every $C_4$-free graph has a subdivision of a clique of order linear in its average degree. We show that every $C_6$-free graph has such a subdivision of a large clique. We also prove the dense case of Mader's conjecture in a stronger sense, i.e. for every $c$, there is a $c'$ such that every $C_4$-free graph with average degree $cn^{1/2}$ has a subdivision of a clique $K_\ell$ with $\ell=\lfloor c'n^{1/2}\rfloor$ where every edge is subdivided exactly $3$ times., 17 pages

Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal sum-free sets in $\{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, S\'os and Temkin on the structure of sum-free sets., Comment: 10 pages, to appear in the Proceedings of the American Mathematical Society

Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, S\'os and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups., Comment: 25 pages, to appear in the Journal of the European Mathematical Society