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1. Coloring graphs with no induced five‐vertex path or gem

   https://doi.org/10.1002/jgt.22572  

   Chudnovsky, Maria; Karthick, T.; Maceli, Peter; Maffray, Frédéric (April 2020, Journal of Graph Theory)

   For a graph 𝐺 , let 𝜒(𝐺) and 𝜔(𝐺) , respectively, denote the chromatic number and clique number of 𝐺 . We give an explicit structural description of ( 𝑃5 , gem)‐free graphs, and show that every such graph 𝐺 satisfies 𝜒(𝐺)≤⌈5𝜔(𝐺)4⌉ . Moreover, this bound is best possible. Here a gem is the graph that consists of an induced four‐vertex path plus a vertex which is adjacent to all the vertices of that path. 

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2. Classifying primitive solvable permutation groups of rank 5 and 6

   https://doi.org/10.1515/jgth-2023-0205  

   Dey, Anakin; O’Neal, Kolton; Tran, Duc_Van Khanh; Upshur, Camron; Yang, Yong (April 2024, Journal of Group Theory)

   Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω.Let $G_0$G_{0}be the stabilizer of a point 𝛼 in Ω.The rank of 𝐺 is defined as the number of orbits of $G_0$G_{0}in Ω, including the trivial orbit $\left\{\alpha \right\}$\{\alpha\}.In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower. 

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3. Actions of 𝐴𝑙𝑡(𝑛) on groups of finite Morley rank without involutions

   https://doi.org/10.1090/proc/16530  

   Altınel, Tuna; Wiscons, Joshua (January 2024, Proceedings of the American Mathematical Society)

   We investigate faithful representations of Alt(n) as automorphisms of a connected group 𝐺 of finite Morley rank. We target a lower bound of 𝑛 on the rank of such a nonsolvable 𝐺, and our main result achieves this in the case when 𝐺 is without involutions. In the course of our analysis, we also prove a corresponding bound for solvable 𝐺 by leveraging recent results on the abelian case. We conclude with an application towards establishing natural limits to the degree of generic transitivity for permutation groups of finite Morley rank. 

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4. Realizing finite groups as automizers

   https://doi.org/10.1515/jgth-2022-0145  

   Bayard, Sylvia; Lynd, Justin (February 2024, Journal of Group Theory)

   Abstract It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺.The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park. 

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5. TEMPERED HOMOGENEOUS SPACES IV

   https://doi.org/10.1017/s1474748022000287  

   Benoist, Yves; Kobayashi, Toshiyuki (June 2022, Journal of the Institute of Mathematics of Jussieu)

   Let 𝐺 be a complex semisimple Lie group and 𝘏 a complex closed connected subgroup. Let g and h be their Lie algebras. We prove that the regular representation of 𝐺 in 𝐿²(𝐺/𝘏) is tempered if and only if the orthogonal of h in g contains regular elements by showing simultaneously the equivalence to other striking conditions, such as h has a solvable limit algebra. 

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