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1. The duality resolution at $n=p=2$

   https://doi.org/10.1007/s00209-025-03754-2  

   Beaudry, Agnès; Bobkova, Irina; Henn, Hans-Werner (July 2025, Mathematische Zeitschrift)

   Abstract Working at the prime 2 and chromatic height 2, we construct a finite resolution of the homotopy fixed points of MoravaE-theory with respect to the subgroup$$\mathbb {G}_2^1$$ $G_2^1$of the Morava stabilizer group. This is an upgrade of the finite resolution of the homotopy fixed points ofE-theory with respect to the subgroup$$\mathbb {S}_2^1$$ $S_2^1$constructed in work of Goerss–Henn–Mahowald–Rezk, Beaudry and Bobkova–Goerss. 

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

   Bethea, Candace (October 2023, 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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3. 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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4. 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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5. Metrics on Permutations With the Same Descent Set

   https://doi.org/10.46787/pump.v8i.4157  

   Diaz-Lopez, Alexander; Haymaker, Kathryn; McGarry, Colin; McMahon, Dylan (January 2025, The PUMP Journal of Undergraduate Research)

   A permutation in a finite symmetric group on a set of ordered elements has a descent at the i-th index if the permutation value at the i-th index is greater than the permutation value that follows. The descent set of a permutation is the set of all indices where the permutation has a descent. Each finite symmetric group can be partitioned by descent sets. In this paper we study the Hamming metric and the L-infinity metric on the sets of permutations that share the same descent set for all nonempty descent sets to determine the maximum possible value that these metrics can achieve when restricted to these subsets.  

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