A Survey of Hypergraph Ramsey Problems

Abstract

One formulation of the Erdős-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on {1, 2, . . . , N }, there exists a monochromatic red s-clique or a monochromatic blue increasing path P n with n vertices, provided N > (s − 1)(n − 1). Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound N > 2 c(log n) s−1 . For the tth power P t n of the ordered increasing graph path with n vertices, we prove a near linear bound c n(log n) s−2 which improves the previous bound that applied to a more general class of graphs than P t n due to Conlon-Fox-Lee-Sudakov.

Longest paths in random hypergraphs

“…Related work. The study of j-tight paths (and the corresponding notion of j-tight cycles) has been a central theme in hypergraph theory, with many generalisations of classical graph results, including Dirac-type and Ramsey-type (see [13,15,19] for surveys), as well as Erdős-Gallai-type results [2,9].…”

On edge‐ordered Ramsey numbers

“…Hypergraph Ramsey numbers have also been actively studied (see, e.g., 13, 35). Their existence was initially proved by Ramsey 36, and better bounds were obtained by Erdős and Rado 20.…”

Independent sets in hypergraphs with a forbidden link

“…4 \ e is the 3-graph on four vertices and three edges. This problem has received considerable attention during the half-century since it was posed, see [7,10,22,23]. Mubayi and Suk [22] wrote that it "is a very interesting open problem, as K (3) 4 \ e is, in some sense, the smallest 3-uniform hypergraph whose Ramsey number with a clique is at least exponential.…”