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Exact solutions to the ErdĆs-Rothschild problem
Let k := (k1,...,k2) be a sequence of natural numbers. For a graph G, let F (G;k) denote the number of colourings of the edges of G with colours 1,...,s such that, for every c â {1,...,s}, the edges of colour c contain no clique of order kc. Write F (n; k) to denote the maximum of F (G;k) over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by ErdĆs and Rothschild in 1974. We prove some new exact results for n â â:
(i) A sufficient condition on k which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results.
(ii) Addressing the original question of ErdĆs and Rothschild, in the case k = (3,..., 3) of length 7, the unique extremal graph is the complete balanced 8-partite graph, with colourings coming from Hadamard matrices of order 8.
(iii) In the case k = (k+ 1, k), for which the sufficient condition in (i) does not hold, for 3 †k †10, the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as
possible
Large cliques or cocliques in hypergraphs with forbidden order-size pairs
The well-known ErdĆs-Hajnal conjecture states that for any graph , there exists such that every -vertex graph that contains no induced copy of has a homogeneous set of size at least . We consider a variant of the ErdĆs-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 vertices and edges for any positive and , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case , for every , 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 . In most cases the bounds are essentially tight. We also determine, for all , whether the growth rate is polynomial or polylogarithmic. Some open problems remain
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Design of new algorithms for gene network reconstruction applied to in silico modeling of biomedical data
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
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
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, ErdĆs, 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 âsymmetricâ 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, ErdĆs, 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
Let denote the graph obtained by intersecting
distinct even cycles at a unique
vertex. In this paper, we determine the unique graphs with maximum adjacency
spectral radius among all graphs on vertices that do not contain any
as a subgraph, for sufficiently large. When
one of the constituent even cycles is a , 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 (see [8, 34, 44, 45])
Ramsey numbers of color critical graphs versus large generalized fans
Given two graphs and , the {Ramsey number} is the smallest
positive integer such that every 2-coloring of the edges of
contains either a red or a blue . Let be the
graph obtained from by adding a new vertex connecting
vertices of . Hook and Isaak (2011) defined the {\em star-critical
Ramsey number} as the smallest integer such that every
2-coloring of the edges of contains either a red or
a blue , where . For sufficiently large , Li and
Rousseau~(1996) proved that , Hao, Lin~(2018)
showed that ;
Li and Liu~(2016) proved that , and Li, Li,
and Wang~(2020) showed that . A graph
with is called edge-critical if contains an edge such
that . In this paper, we extend the above results by showing that
for an edge-critical graph with , when ,
and is sufficiently large, and
.Comment: 10 page
Towards standard imsets for maximal ancestral graphs
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
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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