833 research outputs found

    Large cliques or cocliques in hypergraphs with forbidden order-size pairs

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    The well-known ErdƑs-Hajnal conjecture states that for any graph FF, there exists Ï”>0Ï”>0 such that every nn-vertex graph GG that contains no induced copy of FF has a homogeneous set of size at least nÏ”n^Ï”. We consider a variant of the ErdƑs-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on mm vertices and ff edges for any positive mm and 0≀f≀(m2)0≀f≀(m2), then we obtain large homogeneous sets. For triple systems, in the first nontrivial case m=4m=4, for every S⊆0,1,2,3,4S⊆{0,1,2,3,4}, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in SS. In most cases the bounds are essentially tight. We also determine, for all SS, whether the growth rate is polynomial or polylogarithmic. Some open problems remain

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Design of new algorithms for gene network reconstruction applied to in silico modeling of biomedical data

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    Programa de Doctorado en BiotecnologĂ­a, IngenierĂ­a y TecnologĂ­a QuĂ­micaLĂ­nea de InvestigaciĂłn: IngenierĂ­a, Ciencia de Datos y BioinformĂĄticaClave Programa: DBICĂłdigo LĂ­nea: 111The root causes of disease are still poorly understood. The success of current therapies is limited because persistent diseases are frequently treated based on their symptoms rather than the underlying cause of the disease. Therefore, biomedical research is experiencing a technology-driven shift to data-driven holistic approaches to better characterize the molecular mechanisms causing disease. Using omics data as an input, emerging disciplines like network biology attempt to model the relationships between biomolecules. To this effect, gene co- expression networks arise as a promising tool for deciphering the relationships between genes in large transcriptomic datasets. However, because of their low specificity and high false positive rate, they demonstrate a limited capacity to retrieve the disrupted mechanisms that lead to disease onset, progression, and maintenance. Within the context of statistical modeling, we dove deeper into the reconstruction of gene co-expression networks with the specific goal of discovering disease-specific features directly from expression data. Using ensemble techniques, which combine the results of various metrics, we were able to more precisely capture biologically significant relationships between genes. We were able to find de novo potential disease-specific features with the help of prior biological knowledge and the development of new network inference techniques. Through our different approaches, we analyzed large gene sets across multiple samples and used gene expression as a surrogate marker for the inherent biological processes, reconstructing robust gene co-expression networks that are simple to explore. By mining disease-specific gene co-expression networks we come up with a useful framework for identifying new omics-phenotype associations from conditional expression datasets.In this sense, understanding diseases from the perspective of biological network perturbations will improve personalized medicine, impacting rational biomarker discovery, patient stratification and drug design, and ultimately leading to more targeted therapies.Universidad Pablo de Olavide de Sevilla. Departamento de Deporte e InformĂĄtic

    Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

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    Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lov\'asz Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index

    On the use of senders for minimal Ramsey theory

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    This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt. Let q â©Ÿ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq). In 1976, Burr, ErdƑs, and LovĂĄsz determined, for every s, t â©Ÿ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, qâ©Ÿ3, starting with the ‘symmetric’ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, ErdƑs, and LovĂĄsz in 1976 for pairs of complete graphs. Until now, these senders have been shown to exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area

    Spectral Tur\'an problems for intersecting even cycles

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    Let C2k1,2k2,
,2ktC_{2k_1, 2k_2, \ldots, 2k_t} denote the graph obtained by intersecting tt distinct even cycles C2k1,C2k2,
,C2ktC_{2k_1}, C_{2k_2}, \ldots, C_{2k_t} at a unique vertex. In this paper, we determine the unique graphs with maximum adjacency spectral radius among all graphs on nn vertices that do not contain any C2k1,2k2,
,2ktC_{2k_1, 2k_2, \ldots, 2k_t} as a subgraph, for nn sufficiently large. When one of the constituent even cycles is a C4C_4, our results improve upper bounds on the Tur\'an numbers for intersecting even cycles that follow from more general results of F\"{u}redi [20] and Alon, Krivelevich and Sudakov [1]. Our results may be seen as extensions of previous results for spectral Tur\'an problems on forbidden even cycles C2k,k≄2C_{2k}, k\ge 2 (see [8, 34, 44, 45])

    Ramsey numbers of color critical graphs versus large generalized fans

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    Given two graphs GG and HH, the {Ramsey number} R(G,H)R(G,H) is the smallest positive integer NN such that every 2-coloring of the edges of KNK_{N} contains either a red GG or a blue HH. Let KN−1⊔K1,kK_{N-1}\sqcup K_{1,k} be the graph obtained from KN−1K_{N-1} by adding a new vertex vv connecting kk vertices of KN−1K_{N-1}. Hook and Isaak (2011) defined the {\em star-critical Ramsey number} r∗(G,H)r_{*}(G,H) as the smallest integer kk such that every 2-coloring of the edges of KN−1⊔K1,kK_{N-1}\sqcup K_{1,k} contains either a red GG or a blue HH, where N=R(G,H)N=R(G, H). For sufficiently large nn, Li and Rousseau~(1996) proved that R(Kk+1,K1+nKt)=knt+1R(K_{k+1},K_{1}+nK_{t})=knt +1, Hao, Lin~(2018) showed that r∗(Kk+1,K1+nKt)=(k−1)tn+tr_{*}(K_{k+1},K_{1}+nK_{t})=(k-1)tn+t; Li and Liu~(2016) proved that R(C2k+1,K1+nKt)=2nt+1R(C_{2k+1}, K_{1}+nK_{t})=2nt+1, and Li, Li, and Wang~(2020) showed that r∗(C2m+1,K1+nKt)=nt+tr_{*}(C_{2m+1},K_{1}+nK_{t})=nt+t. A graph GG with χ(G)=k+1\chi(G)=k+1 is called edge-critical if GG contains an edge ee such that χ(G−e)=k\chi(G-e)=k. In this paper, we extend the above results by showing that for an edge-critical graph GG with χ(G)=k+1\chi(G)=k+1, when k≄2k\geq 2, t≄2t\geq 2 and nn is sufficiently large, R(G,K1+nKt)=knt+1R(G, K_{1}+nK_{t})=knt+1 and r∗(G,K1+nKt)=(k−1)nt+tr_{*}(G,K_{1}+nK_{t})=(k-1)nt+t.Comment: 10 page

    Towards standard imsets for maximal ancestral graphs

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    The imsets of Studen\'y (2005) are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice for working with directed acyclic graph (DAG) models. In particular, the 'standard' imset for a DAG is in one-to-one correspondence with the independences it induces, and hence is a label for its Markov equivalence class. We first present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of Hu and Evans (2020). In these cases the imset provides a scoring criteria by measuring the discrepancy for a list of independences that define the model; this gives an alternative to the usual BIC score that is also consistent, and much easier to compute. We also show that, of independence models that do represent the MAG, the imset we give is minimal. Unfortunately, for some graphs the representation does not represent all the independences in the model, and in certain cases does not represent any at all. For these general MAGs, we refine the reduced ordered local Markov property Richardson (2003) by a novel graphical tool called _power DAGs_, and this results in an imset that induces the correct model and which, under a mild condition, can be constructed in polynomial time.Comment: Accepted to Bernoulli, 58 pages, 17 figure

    Water and Brain Function: Effects of Hydration Status on Neurostimulation and Neurorecording

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    Introduction: TMS and EEG are used to study normal neurophysiology, diagnose, and treat clinical neuropsychiatric conditions, but can produce variable results or fail. Both techniques depend on electrical volume conduction, and thus brain volumes. Hydration status can affect brain volumes and functions (including cognition), but effects on these techniques are unknown. We aimed to characterize the effects of hydration on TMS, EEG, and cognitive tasks. Methods: EEG and EMG were recorded during single-pulse TMS, paired-pulse TMS, and cognitive tasks from 32 human participants on dehydrated (12-hour fast/thirst) and rehydrated (1 Liter oral water ingestion in 1 hour) testing days. Hydration status was confirmed with urinalysis. MEP, ERP, and network analyses were performed to examine responses at the muscle, brain, and higher-order functioning. Results: Rehydration decreased motor threshold (increased excitability) and shifted the motor hotspot. Significant effects on TMS measures occurred despite being re-localized and re-dosed to these new parameters. Rehydration increased SICF of the MEP, magnitudes of specific TEP peaks in inhibitory protocols, specific ERP peak magnitudes and reaction time during the cognitive task. Rehydration amplified nodal inhibition around the stimulation site in inhibitory paired-pulse networks and strengthened nodes outside the stimulation site in excitatory and CSP networks. Cognitive performance was not improved by rehydration, although similar performance was achieved with generally weaker network activity. Discussion: Results highlight differences between mild dehydration and rehydration. The rehydrated brain was easier to stimulate with TMS and produced larger responses to external and internal stimuli. This is explainable by the known physiology of body water dynamics, which encompass macroscopic and microscopic volume changes. Rehydration can shift 3D cortical positioning, decrease scalp cortex distance (bringing cortex closer to stimulator/recording electrodes), and cause astrocyte swelling-induced glutamate release. Conclusions: Previously unaccounted variables like osmolarity, astrocyte and brain volumes likely affect neurostimulation/neurorecording. Controlling for and carefully manipulating hydration may reduce variability and improve therapeutic outcomes of neurostimulation. Dehydration is common and produces less excitable circuits. Rehydration should offer a mechanism to macroscopically bring target cortical areas closer to an externally applied neurostimulation device to recruit greater volumes of tissue and microscopically favor excitability in the stimulated circuits
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