Monotone circuit complexity of computing functions on sparse inputs 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.
specifically considering one part of the question (eg for $k_1$=1,$k_2$=2), Lokam studied "2-slice" functions in this paper &amp; proves that strong lower bounds on them can be generalized, therefore this is a very hard open problem related to basic complexity class separation &amp; any such construction/explicit function would be a breakthrough; from the abstract:


  A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1’s and evaluates to one on inputs with more than two 1’s. On inputs with exactly two 1’s f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of 2-slice functions would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. 



Graph Complexity and Slice Functions / Satyanarayana V. Lokam,
Theory Comput. Systems 36, 71–88 (2003)


also as in his comments SJ covers this similar case in his book in the section  exploring star complexity of graphs sec1.7.2.


Boolean function complexity: advances and frontiers, Stasys Jukna