Abstract
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.
Highlights
1.1 Asymptotic induced matchingsWe study in this paper an asymptotic parameter of k-partite k-uniform hypergraphs: the asymptotic induced matching number
We say a subset Ψ of Φ is induced if Ψ = Φ ∩ (Ψ1 × · · · × Ψk) where for each i ∈ [k] we define the marginal set Ψi := {ai : a ∈ Ψ}
We call Ψ a matching if any two distinct elements a, b ∈ Ψ are distinct in all k coordinates, that is, ∀i ∈ [k] : ai = bi
Summary
We study in this paper an asymptotic parameter of k-partite k-uniform hypergraphs: the asymptotic induced matching number. The subrank 1 or induced matching number Q(Φ) is defined as the size of the largest subset Ψ of Φ that is an induced matching, that is, Q(Φ) := max{|Ψ| : Ψ ⊆ Φ, Ψ = Φ ∩ (Ψ1 × · · · × Ψk), ∀a = b ∈ Ψ ∀i ∈ [k] ai = bi}. The asymptotic subrank or the asymptotic induced matching number of the k-graph Φ is defined as. This limit exists and equals the supremum supn∈N Q(Φ n)1/n by Fekete’s lemma [25]. Is a matching, has size n2−o(1) |E| o(n2) when n goes to infinity [27], see [2, Equation 2]
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