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1. An enriched count of nodal orbits in an invariant pencil of conics

   Bethea, Candace (June 2024, arXiv pre-print repository)

   This work gives an equivariantly enriched count of nodal orbits in a general pencil of plane conics that is invariant under a linear action of a finite group on CP2. This can be thought of as spearheading equivariant enumerative enrichments valued in the Burnside Ring, both inspired by and a departure from R(G)-valued enrichments such as Roberts’ equivariant Milnor number and Damon’s equivariant signature formula. Given a G-invariant general pencil of conics, the weighted sum of nodal orbits in the pencil is a formula in terms of the base locus considered as a G-set. We show this is true for all finite groups except Z/2 × Z/2 and D8 and give counterexamples for the two exceptional groups. 

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2. The spread of a finite group

   https://doi.org/doi.org/10.4007/annals.2021.193.2.5  

   Burness, Tim; Guralnick, Robert; Harper, Scott (January 2021, Annals of mathematics)

   null (Ed.)

   A group G is said to be 3/2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property, then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ϵ G, there exists y ϵ G such that G = ⟨x1, y⟩ = ⟨x2, y⟩. In other words, s(G) ⩾ 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods, and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type, and this is the case we treat in this paper. 

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3. Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

   https://doi.org/10.1007/s00493-023-00015-w  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (October 2023, Combinatorica)

   Abstract A graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$ $P_5$-free graph with clique number$$\omega \ge 3$$ $\omega \ge 3$has chromatic number at most$$\omega ^{\log _2(\omega )}$$ ${\omega }^{{log}_2\left(\omega \right)}$. The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$ $(5/27)3^{\omega }$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$ $P_5$, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$ $P_5$-free graphs, and our result is an attempt to approach that. 

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4. Sums of proper divisors follow the Erdős–Kac law

   https://doi.org/10.1090/proc/16167  

   Pollack, Paul; Troupe, Lee (March 2023, Proceedings of the American Mathematical Society)

   Let s ( n ) = ∑ d ∣ n ,   d > n d s(n)=\sum _{d\mid n,~d>n} d denote the sum of the proper divisors of n n . The second-named author proved that ω ( s ( n ) ) \omega (s(n)) has normal order log ⁡ log ⁡ n \log \log {n} , the analogue for s s -values of a classical result of Hardy and Ramanujan [ The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92], AMS Chelsea Publ., Providence, RI, 2000, pp. 262–275]. We establish the corresponding Erdős–Kac theorem: ω ( s ( n ) ) \omega (s(n)) is asymptotically normally distributed with mean and variance log ⁡ log ⁡ n \log \log {n} . The same method applies with s ( n ) s(n) replaced by any of several other unconventional arithmetic functions, such as β ( n ) ≔ ∑ p ∣ n p \beta (n)≔\sum _{p\mid n} p , n − φ ( n ) n-\varphi (n) , and n + τ ( n ) n+\tau (n) ( τ \tau being the divisor function). 

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5. 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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