Sparse halves in dense triangle-free graphs

Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\frac{5}{14}n$ and for graphs with average degree at least $(\frac{2}{5} - \varepsilon)n$ for some absolute $\varepsilon >0$. 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 $1/2$-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 $7/12$-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 $\epsilon>0$, there is a constant $N_{\epsilon}$ (independent of the number $n$ of voters) such that, if the maximum number of votes of the most popular agent is at least $N_{\epsilon}$ then the permutation mechanism is $(\frac{3}{4}-\epsilon)$-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 $(1-\epsilon)$-optimal, for any $\epsilon>0$, provided that the maximum number of votes of the most popular agent is at least a constant $M_{\epsilon}$