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1. Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

   https://doi.org/10.1007/s11856-024-2613-1  

   Hung, Nguyen Ngoc; Sambale, Benjamin; Tiep, Pham Huu (September 2024, Israel Journal of Mathematics)

   Abstract Letk(B₀) andl(B₀) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB₀of a finite groupG. We prove that, ifk(B₀)−l(B₀) = 1, thenl(B₀) ≥p− 1 or elsep= 11 andl(B₀) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B₀) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ $G/O_{p^{\prime }}\left(G\right)\cong C_{11}^2⋊\text{SL}(2,5)$. These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ $2\sqrt{p-1}+1-k_p\left(G\right)$, wherek_p(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups. 

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2. A supplement to Chebotarev’s density theorem

   https://doi.org/10.1007/s11425-022-2141-1  

   Harcos, Gergely; Soundararajan, Kannan (December 2023, Science China Mathematics)

   Abstract LetL/Kbe a Galois extension of number fields with Galois groupG. We show that if the density of prime ideals inKthat split totally inLtends to 1/∣G∣ with a power saving error term, then the density of prime ideals inKwhose Frobenius is a given conjugacy classC⊂Gtends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros ofζ_L(s)/ζ_K(s). 

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3. The growth rate of multicolor Ramsey numbers of 3-graphs

   https://doi.org/10.1007/s40687-024-00463-w  

   Bradač, Domagoj; Fox, Jacob; Sudakov, Benny (September 2024, Research in the Mathematical Sciences)

   Abstract Theq-color Ramsey number of ak-uniform hypergraphG,  denotedr(G; q), is the minimum integerNsuch that any coloring of the edges of the completek-uniform hypergraph onNvertices contains a monochromatic copy ofG. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior ofr(G; q) for fixedGandqtending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height ofr(G; q) as a function ofq. More precisely, given a hypergraphG, we determine whenr(G; q) behaves polynomially, exponentially or double exponentially inq. This answers a question of Axenovich, Gyárfás, Liu and Mubayi. 

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4. Domination inequalities and dominating graphs

   https://doi.org/10.1017/S0305004124000185  

   CONLON, DAVID; LEE, JOONKYUNG (July 2024, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We say that a graphHdominates another graphH^′if the number of homomorphisms fromH^′to any graphGis dominated, in an appropriate sense, by the number of homomorphisms fromHtoG. We study the family of dominating graphs, those graphs with the property that they dominate all of their subgraphs. It has long been known that even-length paths are dominating in this sense and a result of Hatami implies that all weakly norming graphs are dominating. In a previous paper, we showed that every finite reflection group gives rise to a family of weakly norming, and hence dominating, graphs. Here we revisit this connection to show that there is a much broader class of dominating graphs. 

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5. New anomaly free supergravities in six dimensions

   https://doi.org/10.1007/JHEP05(2024)144  

   Becker, K; Kehagias, A; Sezgin, E; Tennyson, D; Violaris, A (May 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> An extended search for anomaly free matter coupledN= (1,0) supergravity in six dimension is carried out by two different methods which we refer to as the graphical and rank methods. In the graphical method the anomaly free models are built from single gauge group models, called nodes, which can only have gravitational anomalies. We search for anomaly free theories with gauge groupsG₁× … ×G_nwithn= 1,2,… (any number of factors) andG₁× … ×G_n×U(1)_Rwheren= 1,2,3 andU(1)_Ris theR-symmetry group. While we primarily consider models with the tensor multiplet numbern_T= 1, we also provide some results forn_T≠ 1 with an unconstrained number of charged hypermultiplets. We find a large number of ungauged anomaly free theories. However, in the case ofR-symmetry gauged models withn_T= 1, in addition to the three known anomaly free theories withG₁×G₂×U(1)_Rtype symmetry, we find only six new remarkably anomaly free models with symmetry groups of the formG₁×G₂×G₃×U(1)_R. In the case ofn_T= 1 and ungauged models, excluding low rank group factors and considering only low lying representations, we find all anomaly free theories. Remarkably, the number of group factors does not exceed four in this class. The proof of completeness in this case relies on a bound which we establish for a parameter characterizing the difference between the number of non-singlet hypermultiplets and the dimension of the gauge group. 

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