arXiv subjects

Vojtěch Dvořák

Publications and source records attributed to Vojtěch Dvořák.

16 recordsLinked to original sources

arXiv2026

Do LLMs Make More Mistakes If They Do Not Believe the Input Data?

Large language models (LLMs) are prone to hallucinating or misinterpreting facts, which impairs their usability in retrieval-augmented generation or data-to-text systems. We analyse how faithfulness of LLMs to provided context depends on how plausible they perceive the context to be (context-memory conflict). To better identify error patterns, we make use of the increased difficulty of non-English and low-resource language text generation and input data based on local knowledge, only partially captured in models' parametric knowledge. We let the models generate text in English, Czech, Slovak and Upper Sorbian from factual (FA), counterfactual (CFA) and fictional (FI) RDF triples containing local Czech and Slovak data. Contrary to our expectations, we observe only a weak context-memory conflict on the human-annotated sample. For Kimi K3 as an LLM judge, which agrees well with human annotations on the sample, counterfactual inputs receive only slightly lower faithfulness scores than factual ones (-0.05 on a 1-5 scale). We also find that a suboptimal choice of LLM judge would lead to overestimating the strength of the context-memory conflict.

cs.CL↗

arXiv2026

DEMUN: Fast and accurate discovery of music notation in very large collections

Much of written musical heritage is preserved and digitised at memory institutions: libraries, museums, and archives. Owing to their collection structures, sheet music tends to be concentrated in large subsets that are defined as collections of music, with corresponding metadata that makes the music findable. However, when studying musical life as opposed to individual works, relevant documents often lie outside of these specialised collections: in textbooks, newspapers, other periodicals, pamphlets, and other documents with extensive circulation. But these documents are typically not catalogued as musical documents, and though there may be a lot of such documents overall, in large library collections, they are still extremely sparse. Manual discovery is thus unfeasible. Automated discovery requires an extremely low false positive rate in order to be useful, and must also operate quickly. We present DEMUN: a two-stage lightweight detector of music notation with a false positive rate of 0.015 %. In the test scenario, 4 million images of a national-scale library were processed, out of which 1,500 pages with music notation were discovered, suggesting the entire collection may contain up to 20-30,000 unmarked documents of musical life.

cs.DL↗

arXiv2026

Optical Music Recognition for Real-World Manuscripts with Synthetic Data

Optical Music Recognition (OMR) has seen major progress in model design, with end-to-end methods now capable of recognising notation at all levels of complexity. However, the impact of this progress has been limited by the visual domains of available training datasets, which are largely born-digital. Existing large collections of sheet music in libraries and other heritage institutions contain predominantly manuscripts, whose visual domains are highly diverse and different, so existing OMR systems fail when applied in the real world. These institutions are often resource-constrained, so large in-domain datasets cannot be expected. We provide a first baseline on real-world manuscripts with complex piano notation in the resource-constrained scenario. Using fine-grained music notation graph (MuNG) annotations and the Smashcima synthesis tool, we then show that while some direct transcriptions of in-domain data remain essential, domain adaptation using synthetic musical manuscript images brings significant improvement. Furthermore, the symbols used do not need to be in-domain, so the expensive fine-grained annotation can be avoided. We thus bring OMR closer to one of its stated goals: preserving and promoting musical cultural heritage.

cs.CV↗

arXiv2026

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.

cs.CV↗

arXiv2024

The Maker-Breaker percolation game on a random board

The $(m,b)$ Maker-Breaker percolation game on $(\mathbb{Z}^2)_p$, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of $\mathbb{Z}^2$ is removed independently with probability $1-p$. After that, Maker chooses a vertex $v_0$ to protect. Then, in each round Maker and Breaker claim respectively $m$ and $b$ unclaimed edges of $G$. Breaker wins if after the removal of the edges claimed by him the component of $v_0$ becomes finite, and Maker wins if she can indefinitely prevent Breaker from winning. We show that for any $p < 1$, Breaker almost surely has a wining strategy for the $(1,1)$ game on $(\mathbb{Z}^2)_p$. This fully answers a question of Day and Falgas-Ravry, who showed that for $p = 1$ Maker has a winning strategy for the $(1,1)$ game. Further, we show that in the $(2,1)$ game on $(\mathbb{Z}^2)_p$ Maker almost surely has a winning strategy whenever $p > 0.9402$, while Breaker almost surely has a winning strategy whenever $p < 0.5278$. This shows that the threshold value of $p$ above which Maker has a winning strategy for the $(2,1)$ game on $\mathbb{Z}^2$ is non-trivial. In fact, we prove similar results in various settings, including other lattices and biases $(m,b)$. These results extend also to the most general case, which we introduce, where each edge is given to Maker with probability $α$ and to Breaker with probability $β$ before the game starts.

math.PR↗

arXiv2022

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 $\Pr[X \geq 1] \geq 6/64$.

math.CO↗

arXiv2022

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.

math.CO↗

arXiv2021

A New Upper Bound for the Ramsey Number of Fans

A fan $F_n$ is a graph consisting of $n$ triangles, all having precisely one common vertex. Currently, the best known bounds for the Ramsey number $R(F_n)$ are $9n/2-5 \leq R(F_n) \leq 11n/2+6$, obtained by Chen, Yu and Zhao. We improve the upper bound to $31n/6+O(1)$.

math.CO↗

arXiv2021

The Maker-Breaker percolation game on the square lattice

We study the $(m,b)$ Maker-Breaker percolation game on $\mathbb{Z}^2$, introduced by Day and Falgas-Ravry. As our first result, we show that Breaker has a winning strategy for the $(m,b)$-game whenever $b \geq (2-\frac{1}{14} + o(1))m$, breaking the ratio $2$ barrier proved by Day and Falgas-Ravry. Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the $(m,2m)$-game even if he allows Maker to claim $c$ edges before the game starts, for any integer $c$, and that he can moreover win rather fast (as a function of $c$). Finally, we consider the game played on $\mathbb{Z}^2$ after the usual bond percolation process with parameter $p$ was performed. We show that when $p$ is not too much larger than $1/2$, Breaker almost surely has a winning strategy for the $(1,1)$-game, even if Maker is allowed to choose the origin after the board is determined.

math.CO↗

arXiv2021

Waiter-Client Triangle-Factor Game on the Edges of the Complete Graph

Consider the following game played by two players, called Waiter and Client, on the edges of $K_n$ (where $n$ is divisible by $3$). Initially, all the edges are unclaimed. In each round, Waiter picks two yet unclaimed edges. Client then chooses one of these two edges to be added to Waiter's graph and one to be added to Client's graph. Waiter wins if she forces Client to create a $K_3$-factor in Client's graph at some point, while if she does not manage to do that, Client wins. It is not difficult to see that for large enough $n$, Waiter has a winning strategy. The question considered by Clemens et al. is how long the game will last if Waiter aims to win as soon as possible, Client aims to delay her as much as possible, and both players play optimally. Denote this optimal number of rounds by $τ_{WC}(\mathcal{F}_{n,K_3-\text{fac}},1 ) $. Clemens et al. proved that $\frac{13}{12}n \leq τ_{WC}(\mathcal{F}_{n,K_3-\text{fac}},1 ) \leq \frac{7}{6}n+o(n) $, and conjectured that $τ_{WC}(\mathcal{F}_{n,K_3-\text{fac}},1 ) = \frac{7}{6}n+o(n) $. In this note, we verify their conjecture.

math.CO↗

arXiv2020

A Note on Restricted Online Ramsey Numbers of Matchings

The restricted online Ramsey numbers were introduced by Conlon, Fox, Grinshpun and He in 2019. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)$ we have $\tilde{R}_{3}(r K_2;n') = 5r-4$.

math.CO↗

arXiv2020

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.

math.CO↗

arXiv2020

$P_{n}$-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.

math.CO↗

arXiv2020

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 $0.46$.

math.CO↗

arXiv2020

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.

math.CO↗

arXiv2019

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 $(\frac{3}{8}+o(1))n$.

math.CO↗