Problem
Let $\mathcal{X}$ be a finite set, the support of a binary function $f: \mathcal{X} \rightarrow \{0,1\}$ is defined as $supp(f)=\{x\in\mathcal{X}: f(x)=1\}$. For any $k\leq \vert \mathcal{X}\vert$, consider the function class $\mathcal{F}_k=\{f: \vert supp(f)\vert \leq k\}$. Find $VC(\mathcal{F}_k)$.
What I Have Done
The function class $\mathcal{F}_k$ is just all functions that predict no more than $k$ positive labels. In order to find VC dimension of $\mathcal{F}_k$, we need to find the largest restriction of $\mathcal{F}_k$ onto $\mathcal{X}$. Therefore, I have $$ \sum_{i=0}^k \binom{n}{i}=2^{VC(\mathcal{F}_k)} $$
However, I do not know how to compute LHS and it seems that it does not have closed form by this answer.
Could someone help me, thank you in advance.
Denote the function in $\mathcal{F}_k$ whose support is $X \subset \mathcal{X}$ by $f_X$. Now consider any $X \subset \mathcal{X}$ with $|X| \leq k$. For any $Y \subset X$, $supp(f_Y) = Y$. Thus $\mathcal F_k$ shatters $X$, meaning $VC(\mathcal{F}_k) \geq k$. Note that $\mathcal{F}_k$ cannot shatter any subset $X \subset \mathcal X$ of size $\geq k$, since its support covers at max $k$ points. Hence $VC(\mathcal F_k) = k$.
I don't understand your argument; the expression you wrote would be true if $n$ (which is not defined) was replaced by $k$, in which case the equation simplifies to $VC(\mathcal F_k) = k$.