Let H1 and H2 be graphs. A graph G has the constrained Ramsey property for (H1,H2) if every edge-colouring of G contains either a monochromatic copy of H1 or a rainbow copy of H2. Our main result gives a 0-statement for the constrained Ramsey property in G(n,p) whenever H1=K1,k for some k≥3 and H2 is not a forest. Along with previous work of Kohayakawa, Konstadinidis and Mota, this resolves the constrained Ramsey property for all non-trivial cases with the exception of H1=K1,2, which is equivalent to the anti-Ramsey property for H2.For a fixed graph H, we say that G has the anti-Ramsey property for H if any proper edge-colouring of G contains a rainbow copy of H. We show that the 0-statement for the anti-Ramsey problem in G(n,p) can be reduced to a (necessary) colouring statement, and use this to find the threshold for the anti-Ramsey property for some particular families of graphs.

Recently, Dragani\'c, Munh\'a Correia, Sudakov and Yuster showed that every tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive tournament on $k$ vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let $f(k)$ be the smallest integer such that any tournament on at least $f(k)$ vertices must contain a $1$-immersion of a transitive tournament on $k$ vertices. We show that $f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one insists in finding an immersion of a complete directed graph on $k$ vertices then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out-degree at least $Ck$ must contain a $2$-immersion of a complete digraph on $k$ vertices. This is again tight up to the value of $C$ and tight on the order of the paths in the immersion.

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r \in \mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. R\"odl and Ruci\'nski and Friedgut, R\"odl and Schacht proved a random version of Rado's theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$ instead of $\mathbb N$. In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold $\hat p:=\hat p(n)$ such that $$\lim _{n \rightarrow \infty} \mathbb P [ S_{n,p} \text{ is } (A,r)\text{-Rado}]= \begin{cases} 0 &\text{ if } p=o(\hat p); \\ 1 &\text{ if } p=\omega(\hat p). \end{cases}$$ Our main result, which we coin the random Rado lemma, is a general black box to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $S_n:=[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem. Various threshold results for abelian groups are also given.

Given graphs G and H, we say G→rH if every r-colouring of the edges of G contains a monochromatic copy of H. Let H[t] denote the t-blowup of H. The blowup Ramsey number B(G→rH;t) is the minimum n such that G[n]→rH[t]. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given G, H and r such that G→rH, there exist constants a=a(G,H,r) and b=b(H,r) such that for all t∈N, B(G→rH;t)≤abt. They conjectured that there exist some graphs H for which the constant a depending on G is necessary. We prove this conjecture by showing that the statement is true in the case of H being 3-chromatically connected, which in particular includes triangles. On the other hand, perhaps surprisingly, we show that for forests F, there exists an upper bound for B(G→rF;t) which is independent of G.Second, we show that for any r,t∈N, any sufficiently large r-edge coloured complete graph on n vertices with Ω(n2−1/t) edges in each colour contains a member from a certain finite family Ftr of r-edge coloured complete graphs. This answers a conjecture of Bowen, Hansberg, Montejano and Müyesser.

A tournament H is quasirandom-forcing if the following holds for every sequence (G_n) of tournaments of growing orders: if the density of H in G_n converges to the expected density of H in a random tournament, then (G_n) is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing, and Coregliano et al. [Electron. J. Combin. 26 (2019), P1.44] showed that there is also a non-transitive 5-vertex tournament with the property. We show that no additional tournament has this property. This extends the result of Bucic et al. [Combinatorica 41 (2021), 175-208] that the non-transitive tournaments with seven or more vertices do not have this property.

For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G(n,p) \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. %We show that to prove the $0$-statement it suffices to prove a deterministic colouring result, which says that if $G$ is not too dense then $G \not \to (H_1,\dots,H_r)$. We extend upon the many partial results for the $0$-statement, by resolving it for a large number of cases, which in particular includes (but is not limited to) when $r \geq 3$, when $H_2$ is strictly $2$-balanced and not bipartite, or when $H_1$ and $H_2$ have the same $2$-densities.

For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G_{n,p} \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. We reduce the $0$-statement of the Kohayakawa--Kreuter conjecture to a natural deterministic colouring problem and resolve this problem for almost all cases, which in particular includes (but is not limited to) when $H_2$ is strictly $2$-balanced and either has density greater than $2$ or is not bipartite. In addition, we extend our reduction to hypergraphs, proving the colouring problem in almost all cases there as well.

A graph H$$ H $$ is common if the number of monochromatic copies of H$$ H $$ in a 2-edge-coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study the notion of weakly locally common graphs considered by Csoka, Hubai, and Lovasz [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2-edge-coloring. We give a complete analysis of the 12 initial terms in the Taylor series determining the number of monochromatic copies of H$$ H $$ in such perturbations and classify graphs H$$ H $$ based on this analysis into three categories: Graphs of Class I are weakly locally common. Graphs of Class II are not weakly locally common. Graphs of Class III cannot be determined to be weakly locally common or not based on the initial 12 terms. As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common.

We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects - so-called Latinons. Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence. Last, using Keevash's recent results on combinatorial designs, we prove that each Latinon can be approximated by a finite Latin square.

We use the hypergraph container theory of Balogh–Morris–Samotij and Saxton–Thomason to obtain general rectilinear approximations and volume estimates for sequences of bodies closed under certain families of projections. We give a number of applications of our results, including a multicolour generalisation of a theorem of Hatami, Janson and Szegedy on the entropy of graph limits. Finally, we raise a number of questions on geometric and analytic approaches to containers.

A famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring of the positive integers gives rise to a monochromatic solution to the equation Ax=0 . Aigner‐Horev and Person recently stated a conjecture on the probability threshold for the binomial random set [n]p having the asymmetric random Rado property: given partition regular matrices A1,…,Ar (for a fixed r≥2 ), however one r‐colors [n]p , there is always a color i∈[r] such that there is an i‐colored solution to Aix=0 . This generalizes the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner‐Horev and Person proved the 1‐statement of their asymmetric conjecture. In this paper, we resolve the 0‐statement in the case where the Aix=0 correspond to single linear equations. Additionally we close a gap in the original proof of the 0‐statement of the (symmetric) random Rado theorem.

The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighbouring vertices receive disjoint translates. The corresponding gyrochromatic number of a graph always lies between the fractional chromatic number and the circular chromatic number. We investigate basic properties of gyrocolorings. In particular, we construct examples of graphs whose gyrochromatic number is strictly between the fractional chromatic number and the circular chromatic number. We also establish several equivalent definitions of the gyrochromatic number, including a version involving all finite abelian groups.

A probability measure on the subsets of the edge set of a graph G is a 1 ‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p . For , Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all , almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p ⋆ . We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f 1, G ( p ), the infimum over all of the probability that is connected. We determine f 1, G ( p ) exactly when G is a path, a complete graph and a cycle of length at most 5.

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Tur\'an) properties of (hyper)graphs and the integers. In particular: (i) We generalise the random Ramsey theorem of R\"odl and Ruci\'nski by providing a resilience analogue. Our result unifies and generalises several fundamental results in the area including the random version of Tur\'an's theorem due to Conlon and Gowers and Schacht. (ii) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter. (iii) All of the above results in fact hold for uniform hypergraphs. (iv) For a (hyper)graph $H$, we determine, up to an error term in the exponent, the number of $n$-vertex (hyper)graphs $G$ that have the Ramsey property with respect to $H$ (that is, whenever $G$ is $r$-coloured, there is a monochromatic copy of $H$ in $G$). (v) We strengthen the random Rado theorem of Friedgut, R\"odl and Schacht by proving a resilience version of the result. (vi) For partition regular matrices $A$ we determine, up to an error term in the exponent, the number of subsets of $\{1,\dots,n\}$ for which there exists an $r$-colouring which contains no monochromatic solutions to $Ax=0$. Along the way a number of open problems are posed.

. Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris, and Samotij [ J. Amer. Math. Soc. , 28 (2015), pp. 669--709], and independently Saxton and Thomason [ Invent. Math. , 201 (2015), pp. 925--992], developed very general container theorems for independent sets in hypergraphs, both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Tur\'an) properties of (hyper)graphs and the integers. In particular we do the following: (a) We generalize the random Ramsey theorem of R\"odl and Ruci\'nski [ Combinatorics, Paul Erd\H os Is Eighty , Vol. 1, Bolyai Soc. Math. Stud., J\'anos Bolyai Mathematical Society, Budapest, 1993, pp. 317--346; Random Structures Algorithms , 5 (1994), pp. 253--270; J. Amer. Math. Soc. , 8 (1995), pp. 917--942] by providing a resilience analogue. Our result unifies and generalizes several fundamental results in the area including the random version of Tur\'an's theorem due to Conlon and Gowers [ Ann. of Math. , 184 (2016), pp. 367--454] and Schacht [ Ann. of Math. , 184 (2016), pp. 331--363]. (b) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter [ Random Structures Algorithms , 11 (1997), pp. 245--276].

Given an integer-valued matrix $A$ of dimension $\ell \times k$ and an integer-valued vector $b$ of dimension $\ell$, the Maker-Breaker $(A,b)$-game on a set of integers $X$ is the game where Maker and Breaker take turns claiming previously unclaimed integers from $X$, and Maker's aim is to obtain a solution to the system $Ax=b$, whereas Breaker's aim is to prevent this. When $X$ is a random subset of $\{1,\dots,n\}$ where each number is included with probability $p$ independently of all others, we determine the threshold probability $p_0$ for when the game is Maker or Breaker's win, for a large class of matrices and vectors. This class includes but is not limited to all pairs $(A,b)$ for which $Ax=b$ corresponds to a single linear equation. The Maker's win statement also extends to a much wider class of matrices which include those which satisfy Rado's partition theorem.

We build up a limit theory for sequences of Latin squares, which parallels the theory of limits of dense graph sequences. Our limit objects, which we call Latinons, are certain two variable functions whose values are probability distributions on [0, 1]. Left-convergence is defined using densities of k x k subpatterns in finite Latin squares, which extends to Latinons. We also provide counterparts to the cut distance, and prove a counting lemma, and an inverse counting lemma.

Given a linear equation $\mathcal{L}$, a set $A \subseteq [n]$ is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. In this paper we consider the following three general questions: (i) What is the size of the largest $\mathcal{L}$-free subset of $[n]$? (ii) How many $\mathcal{L}$-free subsets of $[n]$ are there? (iii) How many maximal $\mathcal{L}$-free subsets of $[n]$ are there? We completely resolve (i) in the case when $\mathcal{L}$ is the equation $px+qy=z$ for fixed $p,q\in \mathbb N$ where $p\geq 2$. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations $\mathcal{L}$, thereby refining a special case of a result of Green. We also give various bounds on the number of maximal $\mathcal{L}$-free subsets of $[n]$ for three-variable homogeneous linear equations $\mathcal{L}$. For this, we make use of container and removal lemmas of Green.

. Given a linear equation L , a set A ⊆ [ n ] is L -free if A does not contain any ‘non-trivial’ solutions to L . In this paper we consider the following three general questions: (i) What is the size of the largest L -free subset of [ n ]? (ii) How many L -free subsets of [ n ] are there? (iii) How many maximal L -free subsets of [ n ] are there? We completely resolve (i) in the case when L is the equation px + qy = z for fixed p,q ∈ N where p ≥ 2. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L , thereby refining a special case of a result of Green [15]. We also give various bounds on the number of maximal L -free subsets of [ n ] for three-variable homogeneous linear equations L . For this, we make use of container and removal lemmas of Green [15].

Given a linear equation L, a set A⊆[n] is L-free if A does not contain any ‘non-trivial’ solutions to L. In this paper we consider the following three general questions:(i)What is the size of the largest L-free subset of [n]?(ii)How many L-free subsets of [n] are there?(iii)How many maximal L-free subsets of [n] are there? We completely resolve (i) in the case when L is the equation px+qy=z for fixed p,q∈N where p≥2. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L, thereby refining a special case of a result of Green [15]. We also give various bounds on the number of maximal L-free subsets of [n] for three-variable homogeneous linear equations L. For this, we make use of container and removal lemmas of Green [15].