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Vojtěch Dvořák

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math.CO math.PR Computer Vision

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preprint2026arXiv

A Dataset for the Recognition of Historical and Handwritten Music Scores in Western Notation

A large amount of musical heritage has been digitised by memory institutions: libraries, museums, and archives. Nevertheless, the field of Optical Music Recognition (OMR) has struggled with making this music machine-readable, despite advances in deep learning, mostly because no datasets for training systems in realistic conditions were available. The MusiCorpus dataset aims to remedy this situation by providing 1,309 pages of historical sheet music, primarily handwritten, with MusicXML transcriptions and symbol annotations. It is the largest dataset of handwritten music to date and the first dataset containing a realistic and representative sample of musical document collections from memory institutions, suitable for training and evaluating both end-to-end and object detection-based OMR systems and comparing their performance.

preprint2022arXiv

Probability Mass of Rademacher Sums Beyond One Standard Deviation

Let $a_1, \dots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$ , and let $\varepsilon_1, \ldots, \varepsilon_n$ be uniformly random $\pm 1$ signs and $X = \sum_{i=1}^{n} a_i \varepsilon_i$ . It is conjectured that $X = \sum_{i=1}^{n} a_i \varepsilon_i$ has $\Pr[X \geq 1] \geq 7/64$ . The best lower bound so far is $1/20$ , due to Oleszkiewicz. In this paper we improve this to $\Pr[X \geq 1] \geq 6/64$ .

preprint2022arXiv

Waiter-Client Clique-Factor Game

Fix two integers $n, k$ , with $n$ divisible by $k$ , and consider the following game played by two players, Waiter and Client, on the edges of $K_n$ . Starting with all the edges marked as unclaimed, in each round, Waiter picks two yet unclaimed edges. Client then chooses one of these edges to be added to Client's graph, while the other edge is added to Waiter's graph. Waiter wins if she eventually forces Client to create a $K_k$ -factor in Client's graph. If she does not manage to do that, Client wins. For fixed $k$ and large enough $n$ , it can be easily shown that Waiter wins if she plays optimally (in particular, this is an immediate consequence of our result that for such $n$ , Waiter can win quite fast). The question posed by Clemens et al. is how long the game will last if Waiter aims to win as fast as she can, Client tries to delay her as much as he can, and they both play optimally. We denote this optimal number of rounds by $τ_{WC}(\mathcal{F}_{n,K_k-\text{fac}},1 )$ . In the present paper, we obtain the first non-trivial lower bound on this quantity for large $k$ . Together with a simple upper bound following the strategy of Clemens et al., this gives $2^{k/3-o(k)}n \leq τ_{WC}(\mathcal{F}_{n,K_k-\text{fac}},1 ) \leq 2^k\frac{n}{k}+C(k)$ , where $C(k)$ is a constant dependent only on $k$ and the $o(k)$ term is independent of $n$ as well.

preprint2020arXiv

$P_{n}$ -induced-saturated graphs exist for all $n \geq 6$

Let $P_{n}$ be a path graph on $n$ vertices. We say that a graph $G$ is $P_{n}$ -induced-saturated if $G$ contains no induced copy of $P_{n}$ , but deleting any edge of $G$ as well as adding to $G$ any edge of $G^{c}$ creates such a copy. Martin and Smith (2012) showed that there is no $P_{4}$ -induced-saturated graph. On the other hand, there trivially exist $P_{n}$ -induced-saturated graphs for $n=2,3$ . Axenovich and Csikós (2019) ask for which integers $n \geq 5$ do there exist $P_{n}$ -induced-saturated graphs. Räty (2019) constructed such a graph for $n=6$ , and Cho, Choi and Park (2019) later constructed such graphs for all $n=3k$ for $k \geq 2$ . We show by a different construction that $P_{n}$ -induced-saturated graphs exist for all $n \geq 6$ , leaving only the case $n=5$ open.

preprint2020arXiv

Improved Bound for Tomaszewski's Problem

In 1986, Tomaszewski made the following conjecture. Given $n$ real numbers $a_{1},...,a_{n}$ with $\sum_{i=1}^{n}a_{i}^{2}=1$ , then of the $2^{n}$ signed sums $\pm a_{1} \pm ... \pm a_{n}$ , at least half have absolute value at most $1$ . Hendriks and Van Zuijlen (2020) and Boppana (2020) independently proved that a proportion of at least $0.4276$ of these sums has absolute value at most $1$ . Using different techniques, we improve this bound to $0.46$ .

preprint2020arXiv

Radius, Girth and Minimum Degree

Given a connected graph $G$ on $n$ vertices, with minimum degree $δ\geq 2$ and girth at least $g \geq 4$ , what is the maximum radius $r$ this graph can have? Erdős, Pach, Pollack and Tuza established in the triangle-free case ( $g=4$ ) that $r \leq \frac{n-2}δ+12$ , and noted that up to the value of the additive constant, this is tight. We determine the exact value for the triangle-free case. For higher $g$ little is known. We settle the order of $r$ for $g=6,8,12$ and prove an upper bound to the order for general even $g$ . Finally, we show that proving the corresponding lower bound for general even $g$ is equivalent to the Erdős girth conjecture.

preprint2020arXiv

The Eternal Game Chromatic Number of Random Graphs

The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this note, we study the eternal game chromatic number of random graphs. We show that with high probability $χ_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$ , and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \mathbb{N}$ . The upper bound applies for even $n$ and any other value of $p$ as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza.

preprint2019arXiv

A Note on Norine's Antipodal-Colouring Conjecture

Norine's antipodal-colouring conjecture, in a form given by Feder and Subi, asserts that whenever the edges of the discrete cube are 2-coloured there must exist a path between two opposite vertices along which there is at most one colour change. The best bound to date was that there must exist such a path with at most $n/2$ colour changes. Our aim in this note is to improve this upper bound to $(\frac{3}{8}+o(1))n$ .

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Vojtěch Dvořák

Research overview

Is this your work?

8 publications

A Dataset for the Recognition of Historical and Handwritten Music Scores in Western Notation

Probability Mass of Rademacher Sums Beyond One Standard Deviation

Waiter-Client Clique-Factor Game

$P_{n}$ -induced-saturated graphs exist for all $n \geq 6$

Improved Bound for Tomaszewski's Problem

Radius, Girth and Minimum Degree

The Eternal Game Chromatic Number of Random Graphs

A Note on Norine's Antipodal-Colouring Conjecture

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Building this map preview

Vojtěch Dvořák

Research overview

Is this your work?

8 publications

A Dataset for the Recognition of Historical and Handwritten Music Scores in Western Notation

Probability Mass of Rademacher Sums Beyond One Standard Deviation

Waiter-Client Clique-Factor Game

PnP_{n}Pn​-induced-saturated graphs exist for all n≥6n \geq 6n≥6

Improved Bound for Tomaszewski's Problem

Radius, Girth and Minimum Degree

The Eternal Game Chromatic Number of Random Graphs

A Note on Norine's Antipodal-Colouring Conjecture

See the researcher in context

Building this map preview

$P_{n}$ -induced-saturated graphs exist for all $n \geq 6$