A Resolution of Friedgut's Conjecture on Influential Coalitions

2026-09-14

AI Summary

本文解决了Friedgut关于影响性联盟的猜想，通过一种新的编码方法和Hatami的结构定理，证明了在任何字母表大小下，存在一个规模为O(n/√log n)的联盟可以以高概率控制函数输出。

Abstract

We prove that, for every constant $\varepsilon>0$ and every function $f:Σ^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ with probability at least $1-\varepsilon$. The bound is independent of the alphabet size and also holds for monotone Boolean functions on $[0,1]^n$, resolving a conjecture of Friedgut (Combinatorics, Probability and Computing, 2004). Unlike the Boolean cube setting, where Kahn, Kalai, and Linial (FOCS, 1988) give a coalition bound of $O(n/\log n)$, no sublinear bound independent of the alphabet size was previously known. In collective coin flipping, our result gives the first sublinear bound on the number of bad players needed to force a fixed output with probability at least $1-\varepsilon$ in any one-round protocol with independent uniform messages, regardless of the message length. A key ingredient in our proof is an encoding that lets us relate the influence of a function on a product space to the $p$-biased influence of the encoded function. We then rely on a structure theorem of Hatami (Annals of Mathematics, 2012) for functions with small $p$-biased influence to bias the encoded function.

Problem

• influential coalitions
• Friedgut's Conjecture
• collective coin flipping
• sublinear bound

Innovation

• influential coalitions
• p-biased influence
• encoding method