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1. 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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2. Concatenating Bipartite Graphs

   https://doi.org/10.37236/8451  

   Chudnovsky, Maria; Hompe, Patrick; Scott, Alex; Seymour, Paul; Spirkl, Sophie (April 2022, The Electronic Journal of Combinatorics)

   Let $$x,y\in (0,1]$$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $$G$$, where every vertex in $$A$$ has at least $x|B|$ neighbours in $$B$$, and every vertex in $$B$$ has at least $y|C|$ neighbours in $$C$$, and there are no edges between $A,C$. We denote by $$\phi(x,y)$$ the maximum $$z$$ such that, in all such graphs $$G$$, there is a vertex $$v\in C$$ that is joined to at least $z|A|$ vertices in $$A$$ by two-edge paths. This function has some interesting properties: we show, for instance, that $$\phi(x,y)=\phi(y,x)$$ for all $x,y$, and there is a discontinuity in $$\phi(x,x)$$ when $1/x$ is an integer. For $z=1/2, 2/3,1/3,3/4,2/5,3/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $$\phi(x,y)\ge z$$ and the pairs with $$\phi(x,y)1/3$. 

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3. Universal minimal flows of extensions of and by compact groups

   https://doi.org/10.1017/etds.2022.52  

   BARTOŠOVÁ, DANA (August 2023, Ergodic Theory and Dynamical Systems)

   Abstract Every topological group G has, up to isomorphism, a unique minimal G -flow that maps onto every minimal G -flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K . Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $$G/K\ltimes K$$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $$2^{\mathbb {N}}$$ , $$M(\mathbb {Z})$$ , or $$M(\mathbb {Z})\times 2^{\mathbb {N}}.$$ 

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4. Triforce and corners

   https://doi.org/10.1017/s0305004119000173  

   FOX, JACOB; SAH, ASHWIN; SAWHNEY, MEHTAAB; STONER, DAVID; ZHAO, YUFEI (July 2020, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ 4– o (1) but not O ( δ 4 ). Let M ( δ ) be the maximum number such that the following holds: for every ∊ > 0 and $$G = {\mathbb{F}}_2^n$$ with n sufficiently large, if A ⊆ G × G with A ≥ δ | G | 2 , then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” ( x , y ), ( x + d , y ), ( x , y + d ) ∈ A is at least ( M ( δ )–∊)| G | 2 . As a corollary via a recent result of Mandache, we conclude that M ( δ ) = δ 4– o (1) and M ( δ ) = ω ( δ 4 ). On the other hand, for 0 < δ < 1/2 and sufficiently large N , there exists A ⊆ [ N ] 3 with | A | ≥ δN 3 such that for every d ≠ 0, the number of corners ( x , y , z ), ( x + d , y , z ), ( x , y + d , z ), ( x , y , z + d ) ∈ A is at most δ c log(1/ δ ) N 3 . A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3. 

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5. Conditions for a Bigraph to be Super-Cyclic

   https://doi.org/10.37236/9683  

   Kostochka, Alexandr; Lavrov, Mikhail; Luo, Ruth; Zirlin, Dara (January 2021, The Electronic Journal of Combinatorics)

   A hypergraph $$\mathcal H$$ is super-pancyclic if for each $$A \subseteq V(\mathcal H)$$ with $$|A| \geqslant 3$$, $$\mathcal H$$ contains a Berge cycle with base vertex set $$A$$. We present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph $$\mathcal H$$ with $$ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$$. We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $$G$$ such that for each $$A \subseteq X$$ with $$|A| \geqslant 3$$, $$G$$ has a cycle $$C_A$$ such that $$V(C_A)\cap X=A$$. Such graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs. 

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