# A ballot-casting problem

For any set $$X$$ and cardinal $$\kappa$$ let $$[X]^\kappa$$ denote the collection of subsets of $$X$$ with cardinality $$\kappa$$. If $$n$$ is a positive integer, let $$[n]:=\{1,\ldots,n\}$$.

Let $$V$$ and $$K$$ be finite, non-empty sets and let $$b$$ be a positive integer with $$b < |K|$$. For the following, we think of $$V$$ as the set of "voters", and $$K$$ is the set of "candidates" and $$b$$ is the "ballot size". If $$f:V \to [K]^b$$ is a map, we can think of $$f$$ as an instance of "ballot casting", and $$f(v)$$ is the "ballot" of voter $$v\in V$$.

Define the set of voters for $$k\in K$$ by $$\text{vote}(k) = \{v \in V: k\in f(v)\}.$$ Call $$k\in K$$ is popular if $$2\cdot \text{vote}(k) > |V|$$, that is, more than half of the voters $$v\in V$$ have $$k$$ on their ballot $$f(v)$$.

For any integer $$n>1$$ we define the happy ballot size of $$n$$ as the smallest positive integer $$b$$ such that there is a finite nonempty set $$V$$ and a map $$f:V \to [[n]]^b$$ such that ever candidate $$k\in[n]$$ is popular. Denote the happy ballot size of $$n$$ by $$\text{hb}(n)$$.

Question. What is the value of $$\lim \inf_{n\to\infty}\frac{\text{hb}(n)}{n}$$?

• @mathworker21 Presumably $[[n]]^b$ is the notation $[\dots]^b$ applied to $[n]=\{1,\dots,n\}$. – Bruno Le Floch Feb 5 at 12:38