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Abstract A fundamental problem in Ramsey theory is to determine the growth rate in terms of $$n$$ of the Ramsey number $$r(H, K_{n}^{(3)})$$ of a fixed $$3$$-uniform hypergraph $$H$$ versus the complete $$3$$-uniform hypergraph with $$n$$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $$H$$, including links of odd cycles and tight cycles of length not divisible by three, $$r(H, K_{n}^{(3)}) \ge 2^{\Omega _{H}(n \log n)}$$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $$H$$ for which $$r(H, K_{n}^{(3)})$$ is superpolynomial in $$n$$. This provides the first example of a separation between $$r(H,K_{n}^{(3)})$$ and $$r(H,K_{n,n,n}^{(3)})$$, since the latter is known to be polynomial in $$n$$ when $$H$$ is linear. 

Abstract Let be a ‐regular graph on vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model . Assign independently to each vertex of a uniform random number , and an edge of is an edge of if and only if . Addressing a problem of Alon and Wei, we prove that if , then with high probability, for each nonnegative integer , there are vertices of degree in . 

Abstract Theq-colour Ramsey number of ak-uniform hypergraphHis the minimum integerNsuch that anyq-colouring of the completek-uniform hypergraph onNvertices contains a monochromatic copy ofH. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximise the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed$$k \ge 3$$and$$q \ge 2$$we prove that the largest possibleq-colour Ramsey number of ak-uniform hypergraph withmedges is at most$$\mathrm{tw}_k(O(\sqrt{m})),$$where tw denotes the tower function. We also present a construction showing that this bound is tight for$$q \ge 4$$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for$$k \geq 4$$and the lower bound for$$k=3$$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs. 

Abstract Theq-color Ramsey number of ak-uniform hypergraphG,  denotedr(G; q), is the minimum integerNsuch that any coloring of the edges of the completek-uniform hypergraph onNvertices contains a monochromatic copy ofG. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior ofr(G; q) for fixedGandqtending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height ofr(G; q) as a function ofq. More precisely, given a hypergraphG, we determine whenr(G; q) behaves polynomially, exponentially or double exponentially inq. This answers a question of Axenovich, Gyárfás, Liu and Mubayi. 

Abstract Let denote the complete 3‐uniform hypergraph on vertices and the 3‐uniform hypergraph on vertices consisting of all edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off‐diagonal Ramsey number exhibits an unusual intermediate growth rate, namely,for some positive constants and . The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum such that any 2‐edge‐coloring of the Cartesian product contains either a red rectangle or a blue ? 

Abstract For an oriented graph , let denote the size of aminimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets. 

Abstract For a subset$$A$$of an abelian group$$G$$, given its size$$|A|$$, its doubling$$\kappa =|A+A|/|A|$$, and a parameter$$s$$which is small compared to$$|A|$$, we study the size of the largest sumset$$A+A'$$that can be guaranteed for a subset$$A'$$of$$A$$of size at most$$s$$. We show that a subset$$A'\subseteq A$$of size at most$$s$$can be found so that$$|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling$$\kappa$$is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets$$A,B$$of$$\mathbb{F}_p$$of size at most$$\alpha p$$for an appropriate constant$$\alpha \gt 0$$, one only needs three elements$$b_1,b_2,b_3\in B$$to guarantee$$|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$$. Allowing the use of larger subsets$$A'$$, we show that for sets$$A$$of bounded doubling, one only needs a subset$$A'$$with$$o(|A|)$$elements to guarantee that$$A+A'=A+A$$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset. 

Abstract The classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at leastrhas the clique of orderras a minor. Hadwiger's conjecture is an example of a well‐studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph onnvertices of independence numberat mostr. If true Hadwiger's conjecture would imply the existence of a clique minor of order. Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition thatGisH‐free for some bipartite graphHthen one can find a polynomially larger clique minor. This has recently been extended to triangle‐free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graphH, answering a question of Dvořák and Yepremyan. In particular, we show that any‐free graph has a clique minor of order, for some constantdepending only ons. The exponent in this result is tight up to a constant factor in front of theterm. 

Abstract The list Ramsey number , recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list‐coloring variant of the classical Ramsey number. They showed that if is a fixed ‐uniform hypergraph that is not ‐partite and the number of colors goes to infinity, . We prove that if and only if is not ‐partite.