We consider the Turán-type problem of bounding the size of a set $M\subseteq {\mathbb{F}}_2^n$ that does not contain a linear copy of a given fixed set $N\subseteq {\mathbb{F}}_2^k$, where n is large compared to k. An Erdős-Stone type theorem [5] in this setting gives a bound that is tight up to a $o(2^n)$ error term; our first main result gives a stability version of this theorem, showing that such an M that is close in size to the upper bound in [5] is close to the obvious extremal example in the sense of symmetric difference. Our second result shows that the error term in [5] is exactly controlled by the solution to one of a class of ‘sparse’ extremal problems, and gives some examples where the error term can be eliminated completely to give a sharp upper bound on $|M|$.

我们考虑限制集合大小的图兰型问题 $\mathrm{中号}\subseteq {\mathbb{F}}_2^{ñ}$ 不包含给定固定集的线性副本 $ñ\subseteq {\mathbb{F}}_2^{ķ}$，其中n比k大。此设置中的Erdős-Stone型定理[5]给出了一个紧至$Ø（2^{ñ}）$错误项 我们的第一个主要结果给出了该定理的稳定性形式，表明在对称差的意义上，这样一个M的大小与[5]中的上限接近的M接近明显的极值示例。我们的第二个结果表明，[5]中的误差项受一类“稀疏”极值问题的解的精确控制，并给出了一些示例，其中误差项可以完全消除以给出一个清晰的上限。$|\mathrm{中号}|$。