  We consider the Relative-Majority Problem (also known as Plurality), in
which, given a multi-agent system where each agent is initially provided an
input value out of a set of $k$ possible ones, each agent is required to
eventually compute the input value with the highest frequency in the initial
configuration. We consider the problem in the general Population Protocols
model in which, given an underlying undirected connected graph whose nodes
represent the agents, edges are selected by a globally fair scheduler.
  The state complexity that is required for solving the Plurality Problem
(i.e., the minimum number of memory states that each agent needs to have in
order to solve the problem), has been a long-standing open problem. The best
protocol so far for the general multi-valued case requires polynomial memory:
Salehkaleybar et al. (2015) devised a protocol that solves the problem by
employing $O(k 2^k)$ states per agent, and they conjectured their upper bound
to be optimal. On the other hand, under the strong assumption that agents
initially agree on a total ordering of the initial input values, Gasieniec et
al. (2017), provided an elegant logarithmic-memory plurality protocol.
  In this work, we refute Salehkaleybar et al.'s conjecture, by providing a
plurality protocol which employs $O(k^{11})$ states per agent. Central to our
result is an ordering protocol which allows to leverage on the plurality
protocol by Gasieniec et al., of independent interest. We also provide a
$\Omega(k^2)$-state lower bound on the necessary memory to solve the problem,
proving that the Plurality Problem cannot be solved within the mere memory
necessary to encode the output.
