56 research outputs found
Sparse halves in dense triangle-free graphs
Erd\H{o}s conjectured that every triangle-free graph on vertices
contains a set of vertices that spans at most
edges. Krivelevich proved the conjecture for graphs with minimum degree at
least . Keevash and Sudakov improved this result to graphs with
average degree at least . We strengthen these results by showing
that the conjecture holds for graphs with minimum degree at least
and for graphs with average degree at least for some absolute . Moreover, we show that the
conjecture is true for graphs which are close to the Petersen graph in edit
distance.Comment: 23 page
A Near-Optimal Mechanism for Impartial Selection
We examine strategy-proof elections to select a winner amongst a set of
agents, each of whom cares only about winning. This impartial selection problem
was introduced independently by Holzman and Moulin and Alon et al. Fisher and
Klimm showed that the permutation mechanism is impartial and -optimal,
that is, it selects an agent who gains, in expectation, at least half the
number of votes of most popular agent. Furthermore, they showed the mechanism
is -optimal if agents cannot abstain in the election. We show that a
better guarantee is possible, provided the most popular agent receives at least
a large enough, but constant, number of votes. Specifically, we prove that, for
any , there is a constant (independent of the number
of voters) such that, if the maximum number of votes of the most popular
agent is at least then the permutation mechanism is
-optimal. This result is tight.
Furthermore, in our main result, we prove that near-optimal impartial
mechanisms exist. In particular, there is an impartial mechanism that is
-optimal, for any , provided that the maximum number
of votes of the most popular agent is at least a constant
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