A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every triangle-divisible graph on n vertices (for n large enough) with minimum degree at least 0.75 n has a triangle decomposition. In this paper, we prove this conjecture in full. In 2016, Barber, Kühn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree cn for some constant c≥ 0.75, then Nash-Williams' Conjecture holds for any constant c' > c. The previously best-known bound on the fractional relaxation was due to Delcourt and Postle from 2021 with c= 7+√(21)/14≈ 0.82733. This bound on the fractional relaxation has grown in importance over the years as it has been directly tied to bounds for a number of other problems in extremal design theory. This paper consists of three parts. In Part I, our first main result is a proof of the Fractional Nash-Williams' Conjecture: if G is a graph on n vertices with minimum degree at least 3n/4, then G has a fractional triangle decomposition. In Part II, our second main result is a Fractional Stability Theorem for Nash-Williams' Conjecture: if a graph G on n vertices has minimum degree close to 3n/4 but no fractional K_3-decomposition, then G is close (in edit distance) to the join of two n/4-regular graphs each on n/2 vertices. We use this to prove that if a triangle-divisible graph G on n vertices has minimum degree close to 3n/4 but no K_3-decomposition, then G is close (in edit distance) to the join of two n/4-regular graphs each on n/2 vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.
The x ad hoc network (xANET) termed as a family of ad hoc networks, including mobile ad hoc network (MANET), vehicular ad hoc network (VANET), flying ad hoc network (FANET) and satellite ad hoc network (SANET), has found a wide range of applications in providing ubiquitous wireless services. Despite its broad utility, the dynamic nature and lack of a centralized controller pose significant challenges to effective and flexible network management for xANET. Conventional network management protocols face challenges such as scalability, security vulnerabilities, configuration complexity, robustness, and performance resilience. Recent efforts have presented different approaches, focusing on policy-based network management (PBNM) and intent-driven network management (IDNM). However, there is no comprehensive survey to clarify their concepts and classifications. This paper presents a survey of the network management evolution for xANET, covering configuration-based, policy-based and the latest intent-driven approaches. We first introduce the characteristics and applications of xANET. Meanwhile, we investigate the network management concepts and challenges. Moreover, we survey the evolution of management protocols for xANET, including simple network management protocol (SNMP), PBNM, and IDNM. Then, we follow the detailed network management of xANET from configuration-based to policy-based and intent-driven approaches. Through comparative analysis, it is found that IDNM employs a more intelligent management protocol, demonstrating higher efficiency and flexibility in handling complex tasks and dynamic network management. This makes it better suited to addressing the challenges of xANET management. Finally, we summarize the remaining challenges and possible future research directions.
In 2014, Keevash famously proved the existence of -Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, K & uuml;hn, and Osthus conjectured a minimum degree generalization: specifically that minimum -degree at least suffices to guarantee that every sufficiently large -divisible -uniform hypergraph on vertices admits a -decomposition (where is a constant that is allowed to depend on but not on ). The best-known progress on this conjecture is from the second proof of the existence conjecture by Glock, K & uuml;hn, Lo, and Osthus in 2016 who showed that suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of was known due to Barber, K & uuml;hn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that suffices for the fractional relaxation. Combined with the work of Henderson and Postle from 2025, this also shows that such -divisible hypergraphs admit -decompositions.
The integration of artificial intelligence (AI) technologies across all segments of space systems, including the launch, space, ground, and user segments, holds immense potential to revolutionize space exploration, satellite operations, and communication networks. This paper presents a comprehensive overview of AI-powered systems in each segment, highlighting their key functionalities, benefits, and challenges. In the launch segment, AI algorithms can optimize launch vehicle trajectories, predict launch conditions, and facilitate the safety of space missions. Machine learning techniques can enable real-time decision-making and autonomous control during launch operations, improving launch success rates and reducing costs. Within space segment, AI-powered satellites can have enhanced capabilities in autonomous navigation, attitude control, and mission planning. These systems leverage AI algorithms to analyze sensor data, detect anomalies, and autonomously adapt to dynamic space environments, increasing mission resilience and flexibility. In the ground segment, AI-powered systems can facilitate satellite operations, data processing, and communication management. Intelligent ground stations utilize machine learning algorithms to optimize antenna pointing, schedule satellite contacts, and process large volumes of satellite data efficiently, enabling faster and more reliable communication services. Finally, in the user segment, AI technologies can enhance the user experience and enable innovative applications in space science, earth observation, and satellite-based services. AI-powered data analytics platforms can provide users with actionable insights from satellite imagery, sensor data, and telemetry, enabling informed decision-making and driving advancements in various domains. Despite the significant benefits offered by AI-powered systems across all segments, challenges such as data quality, algorithm robustness, and ethical considerations remain critical areas for further research and development. Addressing these challenges will be essential to fully harnessing the potential of AI in advancing space exploration, satellite operations, and space-based applications. Through research findings, and technological advancements, this paper aims to provide insights into the current state-of-the-art and future prospects of AI-powered systems in space, paving the way for continued innovation in space exploration.
