Let $G$ be a finite group and $\lambda \in \text{Irr}(G)$ an irreducible complex character of $G$.
Let $m(\lambda) := \min \{ \vert G : H \vert \mid H \leq G, \lambda\vert\_H \text{ has a linear component}\}.$
Is there a universal constant $c$, independent of $G$ and $\lambda$, such that
$$ \frac{m(\lambda)}{\lambda (1\_G)} \leq c? $$
When we consider only monomial groups such a constant is given by 1.
Is there such a constant when we consider only solvable or simple groups?



Existence of universal bound related to characters
There is no such universal constant for solvable groups, at least. Essentially this is because $m(\chi)$ as defined in the question, is multiplicative. That is, if $G\_1$ and $G\_2$ have coprime orders, and if $\chi\_i\in\mathrm{Irr}(G\_i)$ for $i=1,2$ then $\chi\_1\times\chi\_2\in\mathrm{Irr}(G\_1\times G\_2)$ satisfies $m(\chi\_1\times\chi\_2)=m(\chi\_1)m(\chi\_2).$


This follows because the subgroups of $G\_1\times G\_2$ are just the $H\_1\times H\_2$ where $H\_i\subseteq G\_i,$ and $(\chi\_1\times\chi\_2)\_{H\_1\times H\_2}=(\chi\_1)\_{H\_1}\times (\chi\_2)\_{H\_2}$ contains a linear constituent if and only if $(\chi\_i)\_{H\_i}$ contains a linear constituent for $i=1,2.$ Hence if $c(\chi)=m(\chi)/\chi(1)$ then in the situation above, also $c(\chi\_1\times\chi\_2)=c(\chi\_1)c(\chi\_2).$


For odd primes $p>q$ with $q$ dividing $p+1,$ it is known that there exists a solvable group $G$ of order $|G|=p^3q$ and a primitive $\chi\in\mathrm{Irr}(G)$ of degree $p.$ If $H\subseteq G$ and $\chi\_H$ contains a linear constituent then $|G:H|>p$ since $\chi$ is not monomial, so $|G:H|\ge pq$ and $c(\chi)\ge q$ (in fact equality holds). For any $n\ge 1$ we can find $n$ such groups with pairwise coprime orders, for example as follows: Start with $|G\_1|=5^3\cdot 3,$ and for $n\ge 2,$ choose $q\_n$ odd and prime to $\prod\_{i=1}^{n-1}|G\_i|$ and $p\_n$ with $p\_n=-1$ mod $q\_n$ and $p\_n=1$ mod $\prod\_{i=1}^{n-1} |G\_i|.$ Setting $G=\prod\_1^n G\_i$ then $\chi=\prod\_1^n\chi\_i$ (where $\chi\_i\in\mathrm{Irr}(G\_i)$ is any primitive character of $G\_i$ as mentioned above) has $c(\chi)\ge \prod\_1^n q\_i\ge 3^n.$


Since simple groups can't have pairwise coprime orders this doesn't work there, though I guess that there is no universal bound in that case.


