Given an integer $ c \geq 1 $. To each subset $ A $ of the set $ \{1,2, \ldots ,n\} $, we assign a number $ w(A) $ from the set $ \{1,2, \ldots ,c\} $ such that the following condition is satisfied:

Let $ a(n) $ be the number of such assignments. Calculate $ \lim_{n\to \infty}\sqrt[n]{a(n)} $.
Note: $ \min(x,y) $ is the smaller of the numbers $ x $, $ y $.
Let $ M $ be the set $ \{1,2,\ldots,n\} $, and $ M_i $ be the $(n-1)$-element set obtained from $ M $ by removing the number $ i $:

Given the assignment as mentioned in the problem, let's denote the value $ w(M) $ by $ m $, and the value $ w(M_i) $ by $ f(i) $ (for $ i = 1,2, \ldots ,n $). According to the conditions of the problem, the values $ f(1), \ldots ,f(n) $ and $ m $ are numbers from the set $ \{1,2, \ldots ,c\} $, and for each $ i $, the following equality holds:

This is simply a fancy way of writing the inequality:

Thus, the assignment $ w(\cdot) $ is associated with the function:

From the condition imposed on the assignment $ w(\cdot) $:

it follows by induction that the equality:

holds for any sets $ A_1,\ldots,A_k $ contained in $ M $. (The symbol $ \min(x_1,\ldots,x_k) $ denotes the smallest of the numbers $ x_1,\ldots,x_k $; it can, of course, be the common value of several of these numbers.)
Let $ A $ be any subset of the set $ M $, not identical to the entire set $ M $, and let $ i_1,\ldots,i_k $ be all the numbers in $ M $ that do {\it not} belong to $ A $:

The set $ A $ is then exactly the intersection of the sets $ M_{i_1},\ldots,M_{i_k} $:

Indeed: a number $ i $ belongs simultaneously to all sets $ M_{i_1},\ldots,M_{i_k} $ if and only if it belongs to $ M $ and is different from the numbers $ i_1,\ldots,i_k $ - that is, (in accordance with (3)) if and only if it is an element of the set $ A $.
From the connections (4), (2), (3), the equality follows:

(The symbol $ \min_{i \not \in A}f(i) $ denotes the minimum value taken by the function $ f $ on those elements of the set $ M $ that do not belong to the set $ A $.)
Conclusion: the assignment $ w(\cdot) $ is fully determined by specifying the value $ m = w(M) $ and the values $ f(i) = w(M_i) $ for $ i = 1,2,\ldots ,n $. In other words, it is determined by choosing the number $ m $ and the function (1); it is then given by the formula:

And conversely: if $ m $ is any positive integer not exceeding $ c $, and if $ f $ is any mapping of the set $ M $ into the set $ \{1,2,\ldots,m\} $, then the formula (5) defines an assignment $ A \mapsto w(A) $ satisfying the condition stated in the problem.
For a fixed $ m $, there are exactly $ m^n $ functions (1). Therefore, the number of admissible assignments (5) is:

We obtain the two-sided estimate $ c^n \leq a(n) \leq c^n \cdot c $; and hence:

As $ n $ approaches infinity, the $ n $-th root of any fixed positive number approaches $ 1 $. Therefore, the expression on the right side of the inequality (6) approaches $ c $, and by the squeeze theorem, $ \displaystyle \lim_{n \to \infty} \sqrt[n]{a(n)} = c $.
The answer is: The sought limit value is the number $ c $.