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Abstract In 1967, Gerencsér and Gyárfás [16] proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of$$K_n$$, there is a monochromatic path on$$\lceil (2n+1)/3\rceil $$vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erdős [2, 3] regarding the Ramsey numbers of trees (settled in [31]), graphs with bounded maximum degree (settled in [5]), and graphs with bounded degeneracy (settled in [23]). In 1993, Erdős and Galvin [13] began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largestdsuch that in every$$2$$-coloring of$$K_{\mathbb {N}}$$there is a monochromatic infinite path with upper density at leastd? Erdős and Galvin showed that$$2/3\leq d\leq 8/9$$, and after a series of recent improvements, this problem was finally solved in [7] where it was shown that$$d={(12+\sqrt {8})}/{17}$$. This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case. 

Abstract A result of Gyárfás says that for every 3‐coloring of the edges of the complete graph , there is a monochromatic component of order at least , and this is best possible when 4 divides . Furthermore, for all and every ‐coloring of the edges of the complete ‐uniform hypergraph , there is a monochromatic component of order at least and this is best possible for all . Recently, Guggiari and Scott and independently Rahimi proved a strengthening of the graph case in the result above which says that the same conclusion holds if is replaced by any graph on vertices with minimum degree at least ; furthermore, this bound on the minimum degree is best possible. We prove a strengthening of the case in the result above which says that the same conclusion holds if is replaced by any ‐uniform hypergraph on vertices with minimum ‐degree at least ; furthermore, this bound on the ‐degree is best possible. 

Abstract A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary$$r$$-colouring of the complete$$k$$-uniform hypergraph$$K_n^k$$when$$k\geq 2$$and$$k\in \{r-1,r\}$$. We prove a result which says that if one replaces$$K_n^k$$in Gyárfás’ theorem by any ‘expansive’$$k$$-uniform hypergraph on$$n$$vertices (that is, a$$k$$-uniform hypergraph$$G$$on$$n$$vertices in which$$e(V_1, \ldots, V_k)\gt 0$$for all disjoint sets$$V_1, \ldots, V_k\subseteq V(G)$$with$$|V_i|\gt \alpha$$for all$$i\in [k]$$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on$$r$$and$$\alpha$$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary$$r$$-partite$$r$$-uniform hypergraph$$H$$with$$n$$edges in which every set of$$k$$edges has a common intersection. In this language, our result says that if one replaces the condition that every set of$$k$$edges has a common intersection with the condition that for every collection of$$k$$disjoint sets$$E_1, \ldots, E_k\subseteq E(H)$$with$$|E_i|\gt \alpha$$, there exists$$(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$$such that$$e_1\cap \cdots \cap e_k\neq \emptyset$$, then the smallest possible maximum degree of$$H$$is essentially the same (within a small error term depending on$$r$$and$$\alpha$$). We prove our results in this dual setting. 

We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$such that $G⊈K$, or (ii) every countably-infinite tournament contains aspanningcopy of $G$. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals $\mathrm{\kappa <\#comment/>}$; however, we are only able to provide a concise characterization in the case when $\mathrm{\kappa <\#comment/>}={\mathrm{\aleph <\#comment/>}}_1$. 

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $$n$$.  This was previously known to be true for graphs with at most $2.5n-7.5$ edges. We also improve on the best-known lower bounds in the $$r$$-color case. 

Ryser's conjecture says that for every $$r$$-partite hypergraph $$H$$ with matching number $$\nu(H)$$, the vertex cover number is at most $$(r-1)\nu(H)$$.  This far-reaching generalization of König's theorem is only known to be true for $$r\leq 3$$, or when $$\nu(H)=1$$ and $$r\leq 5$$.  An equivalent formulation of Ryser's conjecture is that in every $$r$$-edge coloring of a graph $$G$$ with independence number $$\alpha(G)$$, there exists at most $$(r-1)\alpha(G)$$ monochromatic connected subgraphs which cover the vertex set of $$G$$.   We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs.  Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.