More Like this

1. Packing edge disjoint cliques in graphs

   https://doi.org/10.1007/s00493-025-00184-w  

   Balogh, József; Wigal, Michael C (October 2025, Combinatorica)

   Abstract Let$$r \ge 3$$ $r\ge 3$be fixed andGbe ann-vertex graph. A long-standing conjecture of Győri states that if$$e(G) = t_{r-1}(n) + k$$ $e\left(G\right)=t_{r-1}\left(n\right)+k$, where$$t_{r-1}(n)$$ $t_{r-1}\left(n\right)$denotes the number of edges of the Turán graph onnvertices and$$r - 1$$ $r-1$parts, thenGhas at least$$(2 - o(1))k/r$$ $(2-o(1\left)\right)k/r$edge disjointr-cliques. We prove this conjecture. 

   more » « less
2. 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. 

   more » « less
3. Homotopy equivalent boundaries of cube complexes

   https://doi.org/10.1007/s10711-023-00877-w  

   Fernós, Talia; Futer, David; Hagen, Mark (April 2024, Geometriae Dedicata)

   Abstract A finite-dimensional CAT(0) cube complexXis equipped with several well-studied boundaries. These include theTits boundary$$\partial _TX$$ ${\partial }_{T}X$(which depends on the CAT(0) metric), theRoller boundary$${\partial _R}X$$ ${\partial }_{R}X$(which depends only on the combinatorial structure), and thesimplicial boundary$$\partial _\triangle X$$ ${\partial }_{▵}X$(which also depends only on the combinatorial structure). We use a partial order on a certain quotient of$${\partial _R}X$$ ${\partial }_{R}X$to define a simplicial Roller boundary$${\mathfrak {R}}_\triangle X$$ $R_{▵}X$. Then, we show that$$\partial _TX$$ ${\partial }_{T}X$,$$\partial _\triangle X$$ ${\partial }_{▵}X$, and$${\mathfrak {R}}_\triangle X$$ $R_{▵}X$are all homotopy equivalent,$$\text {Aut}(X)$$ $\text{Aut}\left(X\right)$-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes. 

   more » « less
4. 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. 

   more » « less
5. Sunflowers in Set Systems with Small VC-Dimension

   https://doi.org/10.1007/s00493-025-00186-8  

   Balogh, József; Bernshteyn, Anton; Delcourt, Michelle; Ferber, Asaf; Pham, Huy Tuan (December 2025, Combinatorica)

   Abstract A family ofrdistinct sets$$\{A_1,\ldots , A_r\}$$ $\{A_1,\dots ,A_r\}$is anr-sunflower if for all$$1 \leqslant i < j\leqslant r$$ $1⩽i<j⩽r$and$$1 \leqslant i' < j'\leqslant r$$ $1⩽i^{\prime }<j^{\prime }⩽r$, we have$$A_i\cap A_j = A_{i'}\cap A_{j'}$$ $A_i\cap A_j=A_{i^{\prime }}\cap A_{j^{\prime }}$. Erdős and Rado conjectured in 1960 that every family$$\mathcal {H}$$ $H$of$$\ell $$ $\ell$-element sets of size at least$$K(r)^\ell $$ $K{\left(r\right)}^{\ell }$contains anr-sunflower, whereK(r) is some function that depends only onr. We prove that if$$\mathcal {H}$$ $H$is a family of$$\ell $$ $\ell$-element sets of VC-dimension at mostdand$$|\mathcal H| > (C r(\log d+\log ^*\ell ))^\ell $$ $\left|H\right|>{\left(Cr(logd+{log}^{\ast }\ell )\right)}^{\ell }$for some absolute constant$$C > 0$$ $C>0$, then$$\mathcal {H}$$ $H$contains anr-sunflower. This improves a recent result of Fox, Pach, and Suk. When$$d=1$$ $d=1$, we obtain a sharp bound, namely that$$|\mathcal H| > (r-1)^\ell $$ ${\left|H\right|>(r-1)}^{\ell }$is sufficient. Along the way, we establish a strengthening of the Kahn–Kalai conjecture for set families of bounded VC-dimension, which is of independent interest. 

   more » « less