Inferring Shannon Capacity of pentagon Can we infer from the fact that the number of independent sets in product of $5$-cycle two times is $5$ and in product of $5$-cycle four times is $25$, that the capacity of pentagon is $\sqrt{5}$? That is does $\alpha(C_5 \boxtimes C_5)=5$ and $\alpha(C_5 \boxtimes C_5\boxtimes C_5 \boxtimes C_5)=25$ imply $\theta(C_5)=\sqrt{5}$? Why? Or why not then?

Here $C_5$ is pengtagon, $\alpha(G)$ is independence number of graph $G$, $\boxtimes$ is strong product and $\theta(G)$  is Shannon capacity of graph $G$.
From the givens you can only deduce that the Shannon capacity is at least $\sqrt{5}$. For example, it could be that $\alpha(C_5^8)^{1/8} &gt; \sqrt{5}$, which doesn't contradict your data.

As an aside, the Shannon capacity is usually denoted by $\Theta$, whereas $\theta$ is the Lovász function, which is an upper bound on the Shannon capacity. From the givens you can only deduce that $\theta(C_5) \geq \sqrt{5}$.