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1. Efficient simplicial replacement of semialgebraic sets

   https://doi.org/10.1017/fms.2023.36  

   Basu, Saugata; Karisani, Negin (January 2023, Forum of Mathematics, Sigma)

   Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set$$S \subset \mathbb {R}^k$$by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$, is semialgebraically homeomorphic toS. In this paper, we consider a weaker version of this question. We prove that for any$$\ell \geq 0$$, there exists an algorithm which takes as input a description of a semialgebraic subset$$S \subset \mathbb {R}^k$$given by a quantifier-free first-order formula$$\phi $$in the language of the reals and produces as output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$is$$\ell $$-equivalent toS. The complexity of our algorithm is bounded by$$(sd)^{k^{O(\ell )}}$$, wheresis the number of polynomials appearing in the formula$$\phi $$, andda bound on their degrees. For fixed$$\ell $$, this bound issingly exponentialink. In particular, since$$\ell $$-equivalence implies that thehomotopy groupsup to dimension$$\ell $$of$$|\Delta |$$are isomorphic to those ofS, we obtain a reduction (having singly exponential complexity) of the problem of computing the first$$\ell $$homotopy groups ofSto the combinatorial problem of computing the first$$\ell $$homotopy groups of a finite simplicial complex of size bounded by$$(sd)^{k^{O(\ell )}}$$. 

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2. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao; Longbrake, Sean; Ma, Jie (February 2023, Combinatorics, Probability and Computing)

   Abstract Given a family$$\mathcal{F}$$of bipartite graphs, theZarankiewicz number$$z(m,n,\mathcal{F})$$is the maximum number of edges in an$$m$$by$$n$$bipartite graph$$G$$that does not contain any member of$$\mathcal{F}$$as a subgraph (such$$G$$is called$$\mathcal{F}$$-free). For$$1\leq \beta \lt \alpha \lt 2$$, a family$$\mathcal{F}$$of bipartite graphs is$$(\alpha,\beta )$$-smoothif for some$$\rho \gt 0$$and every$$m\leq n$$,$$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$$(\alpha,\beta )$$-smooth family$$\mathcal{F}$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$$is bipartite. In this paper, we strengthen their result by showing that for every real$$\delta \gt 0$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\delta n^{\alpha -1}$$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$$\mathcal{F}$$consisting of the single graph$$K_{s,t}$$when$$t\gg s$$. We also prove an analogous result for$$C_{2\ell }$$-free graphs for every$$\ell \geq 2$$, which complements a result of Keevash, Sudakov and Verstraëte. 

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3. Large monochromatic components in expansive hypergraphs

   https://doi.org/10.1017/S096354832400004X  

   Bal, Deepak; DeBiasio, Louis (March 2024, Combinatorics, Probability and Computing)

   Abstract A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary$$r$$-colouring of the complete$$k$$-uniform hypergraph$$K_n^k$$when$$k\geq 2$$and$$k\in \{r-1,r\}$$. We prove a result which says that if one replaces$$K_n^k$$in Gyárfás’ theorem by any ‘expansive’$$k$$-uniform hypergraph on$$n$$vertices (that is, a$$k$$-uniform hypergraph$$G$$on$$n$$vertices in which$$e(V_1, \ldots, V_k)\gt 0$$for all disjoint sets$$V_1, \ldots, V_k\subseteq V(G)$$with$$|V_i|\gt \alpha$$for all$$i\in [k]$$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on$$r$$and$$\alpha$$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary$$r$$-partite$$r$$-uniform hypergraph$$H$$with$$n$$edges in which every set of$$k$$edges has a common intersection. In this language, our result says that if one replaces the condition that every set of$$k$$edges has a common intersection with the condition that for every collection of$$k$$disjoint sets$$E_1, \ldots, E_k\subseteq E(H)$$with$$|E_i|\gt \alpha$$, there exists$$(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$$such that$$e_1\cap \cdots \cap e_k\neq \emptyset$$, then the smallest possible maximum degree of$$H$$is essentially the same (within a small error term depending on$$r$$and$$\alpha$$). We prove our results in this dual setting. 

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4. Degrees of maps and multiscale geometry

   https://doi.org/10.1017/fmp.2023.33  

   Berdnikov, Aleksandr; Guth, Larry; Manin, Fedor (January 2024, Forum of Mathematics, Pi)

   Abstract We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $$X_k$$ is the connected sum of k copies of $$\mathbb CP^2$$for$$k \ge 4$$, then we prove that the maximum degree of an L-Lipschitz self-map of $$X_k$$ is between $$C_1 L^4 (\log L)^{-4}$$ and $$C_2 L^4 (\log L)^{-1/2}$$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $$\sim L^n$$. For formal but nonscalable simply connectedn-manifolds, the maximal degree grows roughly like $$L^n (\log L)^{-\theta (1)}$$. And for nonformal simply connected n-manifolds, the maximal degree is bounded by $$L^\alpha $$ for some $$\alpha < n$$. 

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5. Transversal C_k -factors in subgraphs of the balanced blow-up of C_k

   https://doi.org/10.1017/S0963548322000086  

   Ergemlidze, Beka; Molla, Theodore (November 2022, Combinatorics, Probability and Computing)

   Abstract For a subgraph$$G$$of the blow-up of a graph$$F$$, we let$$\delta ^*(G)$$be the smallest minimum degree over all of the bipartite subgraphs of$$G$$induced by pairs of parts that correspond to edges of$$F$$. Johansson proved that if$$G$$is a spanning subgraph of the blow-up of$$C_3$$with parts of size$$n$$and$$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$$, then$$G$$contains$$n$$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$$G$$is a spanning subgraph of the blow-up of$$C_k$$with parts of size$$n$$and$$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$$, then$$G$$contains$$n$$vertex disjoint copies of$$C_k$$such that each$$C_k$$intersects each of the$$k$$parts exactly once. A similar conjecture was also made by Fischer and the case$$k=3$$was proved for large$$n$$by Magyar and Martin. In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of$$G$$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically. 

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