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- W2949259510 abstract "In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $text{VC}$-dimension called $text{VC}_{ell}$-dimension. Let $mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $mathcal{L}$-property is a class of finite $mathcal{L}$-structures closed under isomorphism and substructures. The emph{speed} of a hereditary $mathcal{L}$-property $mathcal{H}$ is the function which sends $n$ to $|mathcal{H}_n|$, where $mathcal{H}_n$ is the set of elements of $mathcal{H}$ with universe ${1,ldots, n}$. It was previously known there exists a gap between the fastest possible speed of a hereditary $mathcal{L}$-property and all lower speeds, namely between the speeds $2^{Theta(n^r)}$ and $2^{o(n^r)}$. We strengthen this gap by showing that for any hereditary $mathcal{L}$-property $mathcal{H}$, either $|mathcal{H}_n|=2^{Theta(n^r)}$ or there is $epsilon>0$ such that for all large enough $n$, $|mathcal{H}_n|leq 2^{n^{r-epsilon}}$. This improves what was previously known about this gap when $rgeq 3$. Further, we show this gap can be characterized in terms of $text{VC}_{ell}$-dimension, therefore drawing a connection between this finite counting problem and the model theoretic dividing line known as $ell$-dependence." @default.
- W2949259510 created "2019-06-27" @default.
- W2949259510 creator A5024872274 @default.
- W2949259510 date "2017-01-02" @default.
- W2949259510 modified "2023-09-26" @default.
- W2949259510 title "$text{VC}_{ell}$-dimension and the jump to the fastest speed of a hereditary $mathcal{L}$-property" @default.
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- W2949259510 doi "https://doi.org/10.48550/arxiv.1701.00470" @default.
- W2949259510 hasPublicationYear "2017" @default.
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