The Vapnik Chervonenkis dimension is defined by the wikipedia page here for a classification model as:

A classification model $$f$$ with some parameter vector $$\theta$$ is said to shatter a set of data points $$(x_1, \ldots x_n)$$ if, for all assignments of labels to those points, there exists some $$\theta$$ such that the model $$f$$ makes no errors when evaluating that set of data points.

I am trying to understand the second example. Suppose I have a model $$f$$ with only one parameter $$\theta$$, and two points $$\{a,b\}$$ where $$a,b \in \mathbb{R}$$ and $$a. Then there are four possible labellings: $$\{a = 1, b=1\}, \{a = 0, b=1\}, \{a = 1, b=0\}, \{a = 0, b=0\}$$

Is the author saying that I have to choose one value of $$\theta$$ that would correctly classify the $$a,b$$ for each of these scenarios? Or rather that I can choose a different $$\theta$$ for each possible labelling.

For instance in the case $$t < a < b$$ where $$a,b$$ are both labelled $$1$$, then by simply classifying points above $$t$$ as positive, I would correctly classify both examples. However this scheme would fail for the case where $$a = 1, b=0$$.

Any insights appreciated.

Here is good definition of VC-dimension -- https://www.cs.hmc.edu/~yjw/teaching/cs158/lectures/21_VCDimension.pdf

To show that hypothesis class has VC-dimension d in input space $$\chi$$, consider this adversarial "shattering game":
• We choose d points in $$\chi$$ positioned however we want;
• We choose a hypothesis $$h \in H$$ that separates the points;
The VC-dimension of $$H$$ in $$\chi$$ is the maximum $$d$$ we can choose so that we always succeed.