The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to percolate if eventually all vertices are infected. For every $r geq 3$, a sharp condition is given for the minimum degree of a sufficiently large graph that guarantees the existence of a percolating set of size $r$. In the case $r=3$, for $n$ large enough, any graph on $n$ vertices with minimum degree $lfloor n/2 floor +1$ has a percolating set of size $3$ and for $r geq 4$ and $n$ large enough (in terms of $r$), every graph on $n$ vertices with minimum degree $lfloor n/2 floor + (r-3)$ has a percolating set of size $r$. A class of examples are given to show the sharpness of these results.
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机译：$ r $ -neighbour引导过程是顶点状态的更新规则，其中包含至少$ r $$）被感染的“未感染”顶点被感染，并且如果最终所有顶点都据说一组最初感染的顶点被感染了。对于每一种$ r geq 3 $，给出了一个充分大的图形的最小程度的尖锐条件，保证了尺寸尺寸的灰尘尺寸为$ r $的存在。在$ r = 3 $，对于$ n $足够大，$ n $顶点的任何图表最低$ lfloor n / 2 rfloor + 1 $的尺寸为$ 3 $和$ r GEQ 4 $和$ n $足够大（根据$ r $），$ n $顶点的每条图表最低$ lfloor n / 2 rfloor +（r-3）$的尺寸$ r $。给出了一类例子来显示这些结果的锐度。
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