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1. On Pisier Type Theorems

   https://doi.org/10.1007/s00493-024-00115-1  

   Nešetřil, Jaroslav; Rödl, Vojtěch; Sales, Marcelo (July 2024, Combinatorica)

   Abstract For any integer$$h\geqslant 2$$ $h⩾2$, a set of integers$$B=\{b_i\}_{i\in I}$$ $B={\left\{b_i\right\}}_{i\in I}$is a$$B_h$$ $B_h$-set if allh-sums$$b_{i_1}+\ldots +b_{i_h}$$ $b_{i_1}+\dots +b_{i_h}$with$$i_1<\ldots $i_1<\dots <i_h$are distinct. Answering a question of Alon and Erdős [2], for every$$h\geqslant 2$$ $h⩾2$we construct a set of integersXwhich is not a union of finitely many$$B_h$$ $B_h$-sets, yet any finite subset$$Y\subseteq X$$ $Y\subseteq X$contains an$$B_h$$ $B_h$-setZwith$$|Z|\geqslant \varepsilon |Y|$$ $\left|Z\right|⩾\epsilon \left|Y\right|$, where$$\varepsilon :=\varepsilon (h)$$ $\epsilon :=\epsilon \left(h\right)$. We also discuss questions related to a problem of Pisier about the existence of a setAwith similar properties when replacing$$B_h$$ $B_h$-sets by the requirement that all finite sums$$\sum _{j\in J}b_j$$ ${\sum }_{j\in J}{b}_j$are distinct. 

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2. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (January 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$for a finite abelian group $$G_0$$ $G_0$, a subsetEof $$G_0$$ $G_0$, and two finite subsets$$F_1,F_2$$ $F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$for sufficiently large d. If one allows the group$$G_0$$ $G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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3. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven_J; Romero_Acosta, Francisco; et al (March 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$, let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$, where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$. Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

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4. Block mapping class groups and their finiteness properties

   https://doi.org/10.1007/s10711-025-00985-9  

   Aramayona, J; Aroca, J; Cumplido, M; Skipper, R; Wu, X (April 2025, Geometriae Dedicata)

   Abstract Given$$g \in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$, let$$\Sigma _g$$ ${\Sigma }_g$denote the closed surface of genusgwith a Cantor set removed, if$$g<\infty $$ $g<\infty$; or the blooming Cantor tree, when$$g= \infty $$ $g=\infty$. We construct a family$$\mathfrak B(H)$$ $B\left(H\right)$of subgroups of$${{\,\textrm{Map}\,}}(\Sigma _g)$$ $\text{Map}\left({\Sigma }_g\right)$whose elements preserve ablock decompositionof$$\Sigma _g$$ ${\Sigma }_g$, andeventually like actlike an element ofH, whereHis a prescribed subgroup of the mapping class group of the block. The group$$\mathfrak B(H)$$ $B\left(H\right)$surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that$$\mathfrak B(H)$$ $B\left(H\right)$is of type$$F_n$$ $F_n$if and only ifHis. As a consequence, for every$$g\in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$and every$$n\ge 1$$ $n\ge 1$, we construct a subgroup$$G <{{\,\textrm{Map}\,}}(\Sigma _g)$$ $G<\text{Map}\left({\Sigma }_g\right)$that is of type$$F_n$$ $F_n$but not of type$$F_{n+1}$$ $F_{n+1}$, and which contains the mapping class group of every compact surface of genus$$\le g$$ $\le g$and with non-empty boundary. 

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5. Search for exotic decays of the Higgs boson to a pair of pseudoscalars in the $$\upmu \upmu \text{ b } \text{ b } $$ and $$\uptau \uptau \text{ b } \text{ b } $$ final states

   https://doi.org/10.1140/epjc/s10052-024-12727-4  

   Hayrapetyan, A; Tumasyan, A; Adam, W; Andrejkovic, J W; Bergauer, T; Chatterjee, S; Damanakis, K; Dragicevic, M; Del_Valle, A Escalante; Hussain, P S; et al (May 2024, The European Physical Journal C)

   Abstract A search for exotic decays of the Higgs boson ($$\text {H}$$ $\text{H}$) with a mass of 125$$\,\text {Ge}\hspace{-.08em}\text {V}$$ $\text{Ge}\text{V}$to a pair of light pseudoscalars$$\text {a}_{1} $$ ${\text{a}}_1$is performed in final states where one pseudoscalar decays to two$${\textrm{b}}$$ $\text{b}$quarks and the other to a pair of muons or$$\tau $$ $\tau$leptons. A data sample of proton–proton collisions at$$\sqrt{s}=13\,\text {Te}\hspace{-.08em}\text {V} $$ $\sqrt{s}=13\text{Te}\text{V}$corresponding to an integrated luminosity of 138$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$recorded with the CMS detector is analyzed. No statistically significant excess is observed over the standard model backgrounds. Upper limits are set at 95% confidence level ($$\text {CL}$$ $\text{CL}$) on the Higgs boson branching fraction to$$\upmu \upmu \text{ b } \text{ b } $$ $\mu \mu \text{b}\text{b}$and to$$\uptau \uptau \text{ b } \text{ b },$$ $\tau \tau \text{b}\text{b},$via a pair of$$\text {a}_{1} $$ ${\text{a}}_1$s. The limits depend on the pseudoscalar mass$$m_{\text {a}_{1}}$$ $m_{{\text{a}}_1}$and are observed to be in the range (0.17–3.3) $$\times 10^{-4}$$ $\times {10}^{-4}$and (1.7–7.7) $$\times 10^{-2}$$ $\times {10}^{-2}$in the$$\upmu \upmu \text{ b } \text{ b } $$ $\mu \mu \text{b}\text{b}$and$$\uptau \uptau \text{ b } \text{ b } $$ $\tau \tau \text{b}\text{b}$final states, respectively. In the framework of models with two Higgs doublets and a complex scalar singlet (2HDM+S), the results of the two final states are combined to determine upper limits on the branching fraction$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} \rightarrow \ell \ell \text{ b } \text{ b})$$ $B(\text{H}\to {\text{a}}_1{\text{a}}_1\to \ell \ell \text{b}\text{b})$at 95%$$\text {CL}$$ $\text{CL}$, with$$\ell $$ $\ell$being a muon or a$$\uptau $$ $\tau$lepton. For different types of 2HDM+S, upper bounds on the branching fraction$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$ $B(\text{H}\to {\text{a}}_1{\text{a}}_1)$are extracted from the combination of the two channels. In most of the Type II 2HDM+S parameter space,$${\mathcal {B}}(\text {H} \rightarrow \text {a}_{1} \text {a}_{1} )$$ $B(\text{H}\to {\text{a}}_1{\text{a}}_1)$values above 0.23 are excluded at 95%$$\text {CL}$$ $\text{CL}$for$$m_{\text {a}_{1}}$$ $m_{{\text{a}}_1}$values between 15 and 60$$\,\text {Ge}\hspace{-.08em}\text {V}$$ $\text{Ge}\text{V}$. 

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