We prove that every $P_5$-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where $P_t$ denotes the t-vertex path, and for graphs $G,H$, we say G is H-free if no induced subgraph of G is isomorphic to H).

More generally, let us say a class $\mathcal{C}$ of graphs is η-bounded if there exists a function $h:\mathbb{N}\to \mathbb{N}$ such that $\eta (G)\le h(\omega (G))$ for every graph $G\in \mathcal{C}$, where $\eta (G)$ denotes smallest cardinality of a hitting set of all maximum stable sets in G, and $\omega (G)$ is the clique number of G. Also, $\mathcal{C}$ is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial.

We introduce η-boundedness inspired by a question of Alon (asking how large $\eta (G)$ can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,

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  given a graph G, we have $\eta (H)\le \omega (H)$ for every induced subgraph H of G if and only if G is perfect;

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  there are graphs G with both $\eta (G)$ and the girth of G arbitrarily large; and

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  if $\mathcal{C}$ is a hereditary class of graphs which is polynomially η-bounded, then $\mathcal{C}$ satisfies the Erdős-Hajnal conjecture.

Unlike χ-boundedness, the case where H is a path is surprisingly hard. Our main result mentioned at the beginning shows that $P_5$-free graphs are η-bounded. The proof is rather involved compared to the classical “Gyárfás path” argument which establishes, for all t, the χ-boundedness of $P_t$-free graphs. It remains open whether $P_t$-free graphs are η-bounded for $t\ge 6$.

It also remains open whether $P_5$-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for $P_5$-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of $P_5$. We further generalize the case where H is a 1-regular graph on four vertices, showing that H-free graphs are polynomially η-bounded if H is a forest with no vertex of degree more than one and at most four vertices of degree one.

我们证明每一个${磷}_5$-有界团数的自由图包含所有其最大稳定集的一个小的命中集（其中${磷}_t$表示t顶点路径，对于图$G,H$，如果 G 没有诱导子图与H同构，我们就说G是H-free 的。

更一般地说，让我们说一个类$\mathcal{C}$如果存在函数，图的个数是η 有界的$H:\mathbb{氮}\to \mathbb{氮}$这样$\eta （G）\le H（\omega （G））$对于每个图$G\epsilon \mathcal{C}$， 在哪里$\eta （G）$表示G中所有最大稳定集合的命中集合的最小基数，并且$\omega （G）$是G的派系数。还，$\mathcal{C}$如果另外h可以选择为多项式，则称其为多项式 η 有界。

我们引入了η有界，其灵感来自于 Alon 的一个问题（询问有多大）$\eta （G）$可以用于 3 色图G )，并受到与χ有界的许多有意义的相似性的启发，即

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  给定图G，我们有$\eta （H）\le \omega （H）$对于G的每个归纳子图H当且仅当G是完美的；

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  有图G两者都有$\eta （G）$G的周长任意大；和

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  如果$\mathcal{C}$是一类遗传性图，其多项式为η有界，则$\mathcal{C}$满足 Erdős-Hajnal 猜想。

与χ有界性不同， H是路径的情况出人意料地困难。我们在开头提到的主要结果表明${磷}_5$-自由图是η有界的。与经典的“Gyárfás 路径”论证相比，证明相当复杂，该论证为所有 t 建立了χ有界性${磷}_t$-免费图表。是否保持开放状态${磷}_t$-自由图是η有界的$t\ge 6$。

它也保持开放状态是否${磷}_5$-自由图是多项式η有界的，如果为真，则意味着 Erdős-Hajnal 猜想${磷}_5$-免费图表。但我们证明，如果H是${磷}_5$。我们进一步推广H是四个顶点上的 1-正则图的情况，表明如果H是一个没有度数超过 1 的顶点且最多有 4 个 1 度顶点的森林，则H无图是多项式η有界的。