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- W4299702773 abstract "The minrank over a field $mathbb{F}$ of a graph $G$ on the vertex set ${1,2,ldots,n}$ is the minimum possible rank of a matrix $M in mathbb{F}^{n times n}$ such that $M_{i,i} neq 0$ for every $i$, and $M_{i,j}=0$ for every distinct non-adjacent vertices $i$ and $j$ in $G$. For an integer $n$, a graph $H$, and a field $mathbb{F}$, let $g(n,H,mathbb{F})$ denote the maximum possible minrank over $mathbb{F}$ of an $n$-vertex graph whose complement contains no copy of $H$. In this paper we study this quantity for various graphs $H$ and fields $mathbb{F}$. For finite fields, we prove by a probabilistic argument a general lower bound on $g(n,H,mathbb{F})$, which yields a nearly tight bound of $Omega(sqrt{n}/log n)$ for the triangle $H=K_3$. For the real field, we prove by an explicit construction that for every non-bipartite graph $H$, $g(n,H,mathbb{R}) geq n^delta$ for some $delta = delta(H)>0$. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl'ak, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry." @default.
- W4299702773 created "2022-10-02" @default.
- W4299702773 creator A5020923873 @default.
- W4299702773 date "2018-06-02" @default.
- W4299702773 modified "2023-09-26" @default.
- W4299702773 title "On Minrank and Forbidden Subgraphs" @default.
- W4299702773 doi "https://doi.org/10.48550/arxiv.1806.00638" @default.
- W4299702773 hasPublicationYear "2018" @default.
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