The asymptotic induced matching number of hypergraphs: balanced binary strings

Srinivasan Arunachalam¹ ; Péter Vrana² ; Jeroen Zuiddam³

¹Massachusetts Institute of Technology
²Budapest University of Technology and Economics
³Institute for Advanced Study

The electronic journal of combinatorics, Tome 27 (2020) no. 3

Résumé

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith–Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science.Our proof relies on a new combinatorial inequality that may be of independent interest. This inequality concerns how many pairs of Boolean vectors of fixed Hamming weight can have their sum in a fixed subspace.

Détail

DOI : 10.37236/9019

Classification : 05C65
Mots-clés : \(k\)-partite \(k\)-uniform hypergraphs