A communication game related to the sensitivity conjecture

One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that interpolates the function) is bounded above by some fixed power of its sensitivity (which is the maximum vertex degree of the graph defined on the inputs where two inputs are adjacent if they differ in exactly one coordinate and their function values are different). We propose an attack on the sensitivity conjecture in terms of a novel two-player communication game. A lower bound of the form $n^{\Omega(1)}$ on the cost of this game would imply the sensitivity conjecture. To investigate the problem of bounding the cost of the game, three natural (stronger) variants of the question are considered. For two of these variants, protocols are presented that show that the hoped for lower bound does not hold. These protocols satisfy a certain monotonicity property, and (in contrast to the situation for the two variants) we show that the cost of any monotone protocol satisfies a strong lower bound. There is an easy upper bound of $\sqrt{n}$ on the cost of the game. We also improve slightly on this upper bound

An Upper Bound on the GKS Game via Max Bipartite Matching

The sensitivity conjecture is a longstanding conjecture concerning the relationship between the degree and sensitivity of a Boolean function. In 2015, a communication game was formulated by Justin Gilmer, Michal Kouck\'{y}, and Michael Saks to attempt to make progress on this conjecture. Andrew Drucker independently formulated this game. Shortly after the creation of the GKS game, Nisan Szegedy obtained a protocol for the game with a cost of $O(n^{.4732})$. We improve Szegedy's result to a cost of $O(n^{.4696})$ by providing a technique to identify whether a set of codewords can be used as a viable strategy in this game

An improvement of the upper bound for GKS communication game

The GKS game was formulated by Justin Gilmer, Michal Koucky, and Michael Saks in their research of the sensitivity conjecture. Mario Szegedy invented a protocol for the game with the cost of $O(n^{0.4732})$. Then a protocol with the cost of $O(n^{0.4696})$ was obtained by DeVon Ingram who used a bipartite matching. We propose a slight improvement of Ingram's method and design a protocol with cost of $O(n^{0.4693})$