#### Vol. 301, No. 1, 2019

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The “quantum” Turán problem for operator systems

### Nik Weaver

Vol. 301 (2019), No. 1, 335–349
DOI: 10.2140/pjm.2019.301.335
##### Abstract

Let $\mathsc{V}$ be a linear subspace of ${M}_{n}\left(ℂ\right)$ which contains the identity matrix and is stable under Hermitian transpose. A “quantum $k$-clique” for $\mathsc{V}$ is a rank $k$ orthogonal projection $P\in {M}_{n}\left(ℂ\right)$ for which $dim\left(P\mathsc{V}P\right)={k}^{2}$, and a “quantum $k$-anticlique” is a rank $k$ orthogonal projection for which $dim\left(P\mathsc{V}P\right)=1$. We give upper and lower bounds both for the largest dimension of $\mathsc{V}$ which would ensure the existence of a quantum $k$-anticlique, and for the smallest dimension of $\mathsc{V}$ which would ensure the existence of a quantum $k$-clique.

##### Keywords
operator systems, Turán problem, quantum graph theory
##### Mathematical Subject Classification 2010
Primary: 05C69, 05D10, 46L07, 81P45