Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The main result shows that it suffices to consider only 3colorings which have a special partition of the vertices. Using this tool, we find several sharp numbers and conjecture the sharp value for all bipartite graphs. In particular, we determine the Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part and initiate the study of linear forests.

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In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ − → rc(T ) ≤ 3.

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Linkage disequilibrium (LD) is the correlation among nearby genetic variants. In genetic association studies, LD is often modeled using massive local correlation matrices, but this approach is slow, especially in ancestrally diverse studies. Here, we introduce LD graphical models (LDGMs), which are an extremely sparse and efficient representation of LD. LDGMs are derived from genome-wide genealogies; statistical relationships among alleles in the LDGM correspond to genealogical relationships among haplotypes. We publish LDGMs and ancestry specific LDGM precision matrices for 18 million common SNPs (MAF>1%) in five ancestry groups, validate their accuracy, and demonstrate order-of-magnitude improvements in runtime for commonly used LD matrix computations. We implement an extremely fast multi-ancestry polygenic prediction method, BLUPx-ldgm, which performs better than a similar method based on the reference LD correlation matrix. LDGMs will enable sophisticated methods that scale to ancestrally genetic association data across millions of variants and individuals.