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1. Colouring graphs with forbidden bipartite subgraphs

   https://doi.org/10.1017/S0963548322000104  

   Anderson, James; Bernshteyn, Anton; Dhawan, Abhishek (January 2023, Combinatorics, Probability and Computing)

   Abstract A conjecture of Alon, Krivelevich and Sudakov states that, for any graph $$F$$ , there is a constant $$c_F \gt 0$$ such that if $$G$$ is an $$F$$ -free graph of maximum degree $$\Delta$$ , then $$\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$$ . Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs $$F$$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if $$G$$ is $$K_{t,t}$$ -free, then $$\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$$ as $$\Delta \to \infty$$ . We improve this bound to $$(1+o(1)) \Delta/\log\!\Delta$$ , making the constant factor independent of $$t$$ . We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle. 

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2. $$K_{r,s}$$ Graph Bootstrap Percolation

   https://doi.org/10.37236/8997  

   Bayraktar, Erhan; Chakraborty, Suman (January 2022, The Electronic Journal of Combinatorics)

   A graph $$G$$ percolates in the $$K_{r,s}$$-bootstrap process if we can add all missing edges of $$G$$ in some order such that each edge creates a new copy of $$K_{r,s}$$, where $$K_{r,s}$$ is the complete bipartite graph. We study $$K_{r,s}$$-bootstrap percolation on the Erdős-Rényi random graph, and determine the percolation threshold for balanced $$K_{r,s}$$ up to a logarithmic factor. This partially answers a question raised by Balogh, Bollobás, and Morris. We also establish a general lower bound of the percolation threshold for all $$K_{r,s}$$, with $$r\geq s \geq 3$$. 

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3. Purity and Separation for Oriented Matroids

   https://doi.org/10.1090/memo/1439  

   Galashin, Pavel; Postnikov, Alexander (September 2023, Memoirs of the American Mathematical Society)

   Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has the same cardinality. In this paper, we extend these notions and define $$\mathcal{M}$$-separated collections for any oriented matroid $$\mathcal{M}$$. We show that maximal by size $$\mathcal{M}$$-separated collections are in bijection with fine zonotopal tilings (if $$\mathcal{M}$$ is a realizable oriented matroid), or with one-element liftings of $$\mathcal{M}$$ in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid $$\mathcal{M}$$ is pure if $$\mathcal{M}$$-separated collections form a pure simplicial complex, i.e., any maximal by inclusion $$\mathcal{M}$$-separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank $$3$$, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank $$3$$ is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an $$n$$-gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank $$3$$, graphical, uniform). 

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4. THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

   https://doi.org/10.1017/S1474748019000318  

   Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto (March 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

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5. A Unified Proof of Conjectures on Cycle Lengths in Graphs

   https://doi.org/10.1093/imrn/rnaa324  

   Gao, Jun; Huo, Qingyi; Liu, Chun-Hung; Ma, Jie (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach: 1. Every graph $$G$$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $$k$$; in addition, if $$G$$ is $$2$$-connected and non-bipartite, then it contains cycles of all lengths modulo $$k$$. 2. For all $$k\geq 3$$, every $$k$$-connected graph contains a cycle of length zero modulo $$k$$. 3. Every $$3$$-connected non-bipartite graph with minimum degree at least $k+1$ contains $$k$$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $$k$$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible. 

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