The Ramsey number $R_X(p,q)$ for a class of graphs $X$ is the minimum $n$ such that every graph in $X$ with at least $n$ vertices has either a clique of size $p$ or an independent set of size $q$. We say that Ramsey numbers are linear in $X$ if there is a constant $k$ such that $R_X(p,q)\le k(p+q)$ for all $p,q$. In the present paper we conjecture that if $X$ is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in $X$ if and only if $X$ excludes a forest, a disjoint union of cliques and their complements. We prove the “only if” part of this conjecture and verify the “if” part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.

拉姆齐数 ${\mathrm{[R}}_X（p，q）$ 对于一类图 $X$ 是最小的 $ñ$ 这样每个图 $X$ 至少 $ñ$ 顶点具有大小族 $p$ 或一组独立的尺寸 $q$。我们说拉姆齐数是线性的$X$ 如果有一个常数 $ķ$ 这样 ${\mathrm{[R}}_X（p，q）\le ķ（p+q）$ 对全部 $p，q$。在本文中，我们推测$X$ 是由有限个禁止的诱导子图定义的世袭类别，则拉姆齐数在 $X$ 当且仅当 $X$不包括森林，不连贯的派系及其补充。我们证明此猜想的“仅当”部分，并针对各种类别验证“如果”部分。我们还将线性概念应用于二分Ramsey数，并揭示了二分和非二分情况之间的许多相似之处和不同之处。