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Abstract An edge‐coloured graph is said to berainbowif no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge‐coloured graph on vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of for this question. Very recently, Janzer–Sudakov and Kim–Lee–Liu–Tran independently removed the term in Tomon's bound, showing a bound of . We prove an upper bound of for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the term, and so, it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non‐abelian groups. 

ABSTRACT Below-ground carbon transformations that contribute to healthy soils represent a natural climate change mitigation, but newly acquired traits adaptive to climate stress may alter microbial feedback mechanisms. To better define microbial evolutionary responses to long-term climate warming, we study microorganisms from an ongoingin situsoil warming experiment where, for over three decades, temperate forest soils are continuously heated at 5°C above ambient. We hypothesize that across generations of chronic warming, genomic signatures within diverse bacterial lineages reflect adaptations related to growth and carbon utilization. From our bacterial culture collection isolated from experimental heated and control plots, we sequenced genomes representing dominant taxa sensitive to warming, including lineages of Actinobacteria, Alphaproteobacteria, and Betaproteobacteria. We investigated genomic attributes and functional gene content to identify signatures of adaptation. Comparative pangenomics revealed accessory gene clusters related to central metabolism, competition, and carbon substrate degradation, with few functional annotations explicitly associated with long-term warming. Trends in functional gene patterns suggest genomes from heated plots were relatively enriched in central carbohydrate and nitrogen metabolism pathways, while genomes from control plots were relatively enriched in amino acid and fatty acid metabolism pathways. We observed that genomes from heated plots had less codon bias, suggesting potential adaptive traits related to growth or growth efficiency. Codon usage bias varied for organisms with similar 16Srrnoperon copy number, suggesting that these organisms experience different selective pressures on growth efficiency. Our work suggests the emergence of lineage-specific trends as well as common ecological-evolutionary microbial responses to climate change.IMPORTANCEAnthropogenic climate change threatens soil ecosystem health in part by altering below-ground carbon cycling carried out by microbes. Microbial evolutionary responses are often overshadowed by community-level ecological responses, but adaptive responses represent potential changes in traits and functional potential that may alter ecosystem function. We predict that microbes are adapting to climate change stressors like soil warming. To test this, we analyzed the genomes of bacteria from a soil warming experiment where soil plots have been experimentally heated 5°C above ambient for over 30 years. While genomic attributes were unchanged by long-term warming, we observed trends in functional gene content related to carbon and nitrogen usage and genomic indicators of growth efficiency. These responses may represent new parameters in how soil ecosystems feedback to the climate system. 

We observe that several vertex Turán type problems for the hypercube that received a considerable amount of attention in the combinatorial community are equivalent to questions about erasure list-decodable codes. Analyzing a recent construction of Ellis, Ivan and Leader, and determining the Turán density of certain hypergraph augmentations we obtain improved bounds for some of these problems. 

A Simpler Approach to the EFX Problem Envy-freeness up to any item (EFX) has emerged as a compelling fairness notion in discrete fair division. However, its existence remains one of the biggest open problems in the field. In a breakthrough, Chaudhury et al. (2020) establish the existence of EFX allocations for three agents with additive valuations through intricate case analysis. The paper “EFX: A Simpler Approach and an (Almost) Optimal Guarantee via Rainbow Cycle Number” by Akrami, Alon, Chaudhury, Garg, Mehlhorn, and Mehta offers a simpler approach for improving the EFX guarantee. They demonstrate the existence of EFX allocations for three agents when at least one has additive valuations (whereas the other two have general monotone valuations). Additionally, they nearly resolve a conjecture regarding the rainbow cycle number, leading to an (almost) tight bound for the existence of approximate EFX allocations with few unallocated items achievable through this approach. 

ABSTRACT The goal of the MRI4ALL hackathon, which took place in October 2023, was to develop a functional low‐field MRI scanner in just one week and to release all created source code and resources as open‐source packages. The event was attended by 52 participants from 16 institutions who assembled the scanner on the last day of the hackathon. The scanner's magnetic B₀field with a strength of 43 mT and a target field‐of‐view size of 11 cm³was created with a Halbach array made from 990 N40UH permanent magnets, held in place using 3D printed ring formers. Gradient coils were fabricated by gluing enameled copper wire onto 3D printed holders with imprinted wire patterns. A solenoid coil for RF transmission and reception was built by winding 20 turns of Litz wire around a 3D printed cylinder. A Red Pitaya FPGA prototyping board running the MaRCoS framework was used to control the scanner components, and a GPA‐FHDO amplifier board was used to drive the gradients. To simplify the scanner's operation, console software with an intuitive graphical user interface was developed in Python using the PyPulseq package for sequence calculations. Furthermore, the scanner was equipped with a cooling system, as well as options for passive and active shimming. After resolving several technical issues that arose during the assembly, the scanner is now able to acquire MR images with different sequences. While not suitable for real‐world clinical applications, it can be utilized for educational purposes or as a low‐cost prototyping platform. Moreover, it may serve as a reference design for future MRI development projects. All source code and resources are available on the project websitemri4all.org, allowing other groups to replicate the scanner. Evidence Leveln/a Technical EfficacyStage 1.