# What does a kernel of size n,n^2 ,… mean?

So according to Wikipedia,

In the Notation of [Flum and Grohe (2006)], a ”parameterized problem” consists of a decision problem $$L\subseteq\Sigma^*$$ and a function $$\kappa:\Sigma^*\to N$$, the parameterization. The ”parameter” of an instance $$x$$ is the number $$\kappa(x)$$. A ”’kernelization”’ for a parameterized problem $$L$$ is an algorithm that takes an instance $$x$$ with parameter $$k$$ and maps it in polynomial time to an instance $$y$$ such that

• $$x$$ is in $$L$$ if and only if $$y$$ is in $$L$$ and
• the size of $$y$$ is bounded by a computable function $$f$$ in $$k$$. Note that in this notation, the bound on the size of $$y$$ implies that the parameter of $$y$$ is also bounded by a function in $$k$$.

The function $$f$$ is often referred to as the size of the kernel. If $$f=k^{O(1)}$$, it is said that $$L$$ admits a polynomial kernel. Similarly, for $$f={O(k)}$$, the problem admits linear kernel. ”’

Stupid question, but since the parameter can be anything can’t you just define the parameter to be really large and then you always have linear kernel?