The weight $|x|$ of a binary string $x\in\{0,1\}^n$ is the number of ones in the string. What happens if we are interested in computing a monotone function on inputs with few ones?
We know that deciding if a graph has a $k$-clique is hard for monotone circuits (see among others Alon Boppana, 1987), but if a graph has for example at most $k^3$ edges it possible to find a monotone bounded depth circuit of size $f(k)\cdot n^{O(1)}$ which decides $k$-clique.
My question: is there any function which is hard to compute by a monotone circuit even on inputs of weight less than $k$? Here hard means circuit size $n^{{k}^{\Omega(1)}}$.
Even better: is there an explicit monotone function which is hard to compute even if we only care about inputs of weight $k_1$ and $k_2$?
Emil Jeřábek already observed that known lower bounds hold for monotone circuits that separate two classes of inputs ($a$-cliques vs maximal $(a-1)$-colorable graphs), thus at cost of some independence in the probabilistic argument it is possible to make it work for two classes of input of fixed weight. This would cause $k_2$ to be a function of $n$ which I want to avoid.
What is would really like is an explicit hard function for $k_1$ and $k_2$ much smaller than $n$ (as in the parameterized complexity framework). Even better if $k_1=k_2+1$.
Notice that a positive answer for $k_1=k_2$ would imply an exponential lower bound for arbitrary circuits.
Update: This question may be partially relevant.
