Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kolmogorov complexity as a combinatorial tool

Alexander Shen

TL;DR

The paper investigates nuanced uses of Kolmogorov complexity in combinatorics beyond simple counting arguments, introducing a new complexity-based construction for combinatorial rectangles and a corresponding Muchnik-game interpretation. It defines $i(A)=\min_{x\in A} C(A|x)$ and proves the rectangle complexity inequality $C(R) \ge C(R|x) + C(R|y) - O(\log n + i(R))$ for all $R$ and $(x,y)\in R$, via the copy lemma. The rectangle-game framework shows that, for a polynomial bound $p(n)$, a second player can win the game with appropriate parameters, enabling constructive strategies and linking to applications like opinions aggregation (Kozachinskiy--Steifer), while highlighting how additive constants in $C(\cdot)$ affect explicit bounds. The discussion emphasizes the trade-offs between asymptotic existence results and explicit combinatorial bounds, and suggests directions where complexity methods yield tangible, explicit protocols in combinatorics.

Abstract

Kolmogorov complexity is often used as a convenient language for counting and/or probabilistic existence proofs. However, there are some applications where Kolmogorov complexity is used in a more subtle way. We provide one (somehow) surprising example where an existence of a winning strategy in a natural combinatorial game is proven (and no direct proof is known).

Kolmogorov complexity as a combinatorial tool

TL;DR

The paper investigates nuanced uses of Kolmogorov complexity in combinatorics beyond simple counting arguments, introducing a new complexity-based construction for combinatorial rectangles and a corresponding Muchnik-game interpretation. It defines and proves the rectangle complexity inequality for all and , via the copy lemma. The rectangle-game framework shows that, for a polynomial bound , a second player can win the game with appropriate parameters, enabling constructive strategies and linking to applications like opinions aggregation (Kozachinskiy--Steifer), while highlighting how additive constants in affect explicit bounds. The discussion emphasizes the trade-offs between asymptotic existence results and explicit combinatorial bounds, and suggests directions where complexity methods yield tangible, explicit protocols in combinatorics.

Abstract

Kolmogorov complexity is often used as a convenient language for counting and/or probabilistic existence proofs. However, there are some applications where Kolmogorov complexity is used in a more subtle way. We provide one (somehow) surprising example where an existence of a winning strategy in a natural combinatorial game is proven (and no direct proof is known).
Paper Structure (2 sections, 3 theorems, 2 equations)

This paper contains 2 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. New example

Key Result

Theorem 1

for every combinatorial rectangle $R\subset \mathbb{B}^n\times\mathbb{B}^n$ and every its element $(x,y)\in R$.

Theorems & Definitions (3)

  • Theorem 1: Romashchenko, Zimand, lemma 4.6 in rz adapted to one rectangle
  • Theorem 2
  • Theorem 3: Kozachinskiy--Steifer