Sergey Norin

A counterexample to a conjecture of Schwartz

In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition A, B of its vertex set, such that every transitive subset of A is in the out-neighbour set of some vertex in B, and vice versa. But in fact there is such a tournament, as we show in this article, and so Schwartz’ conjecture is false. Our proof is non-constructive and uses the probabilistic method.

Counting flags in triangle-free digraphs

Motivated by the Caccetta-Häggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0:3465n contains an oriented triangle. This improves the bound of 0:3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.

Polylogarithmic Supports Are Required for Approximate Well-Supported Nash Equilibria below 2/3

In an e-approximate Nash equilibrium, a player can gain at most e in expectation by unilateral deviation. An e-well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within e of the best response payoff. Daskalakis, Mehta and Papadimitriou [8] conjectured that every win-lose bimatrix game has a

Turán numbers of extensions

\frac{2}{3}

Strongly Sublinear Separators and Polynomial Expansion

-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a

A Near-Optimal Mechanism for Impartial Selection

\frac23

Large Supports are Required for Well-Supported Nash Equilibria

payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least

The extremal function for disconnected minors

\Omega\sqrt[3]{\log n}

Flag algebras and the stable coefficients of the Jones polynomial

in any e-well supported equilibrium with

Orthogonal Tree Decompositions of Graphs

\epsilon . The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture [4]. A probabilistic argument [13] shows that there exist e-well-supported equilibria with supports of cardinality