A survey of quantitative bounds for hypergraph Ramsey problems

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of ${1,ldots, N}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments. We also include several new results and proofs that have not been published elsewhere.

经典的HyperGraph Ramsey编号$ r_k（s，n）$是最低$ n $，因此对于$ k $ tupples $ {1，ldots，n} $的每种红蓝色颜色，都有$ S $整数使它们中的每个$ k $ tuple是红色或$ n $整数，因此它们中的每个$ k $ tuple是蓝色的。我们调查了各种问题和超图拉姆西理论的结果，这些理论已经不了解$ r_k（s，n）$的定量方面而逐渐成长。我们的重点是最近的发展。我们还包括几个新的结果和证据，这些结果和证据尚未在其他地方发布。