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Let X be a simplicial complex on vertex set V. We say that X is d-representable if it is isomorphic to the nerve of a family of convex sets in R d . We define the d-boxicity of X as the minimal k such that X can be written as the intersection of kd-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of X is a set τ ⊂ V such that τ ∉ X but σ ∈ X for any σ ⊊ τ . We prove that the d-boxicity of a simplicial complex on n vertices without missing faces of dimension larger than d is at most ⌊ n d / (d + 1) ⌋ . The bound is sharp: the d-boxicity of a simplicial complex whose set of missing faces form a Steiner (d , d + 1 , n) -system is exactly n d / (d + 1) . One of the main ingredients in the proof is the following bound on the representability of a complex: let V 1 , ... , V k be subsets of V such that V i ∉ X for all 1 ≤ i ≤ k , and for any missing face τ of X there is some 1 ≤ i ≤ k satisfying | τ \ V i | ≤ 1 . Then, X can be written as an intersection X = ⋂ i = 1 k X i , where, for all 1 ≤ i ≤ k , X i is a (| V i | - 1) -representable complex. In particular, X is (∑ i = 1 k (| V i | - 1)) -representable. [ABSTRACT FROM AUTHOR]

The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov (On growth rate in S L 2 (F p) , the affine group and sum-product type implications, 2018. arXiv:1812.01671). We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds in Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound from Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite A ⊂ R the following superquadratic bound holds: This improves on a bound with exponent 2 that was given in Rudnev and Shkredov. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant c such that for all finite A , B ⊂ R , [ABSTRACT FROM AUTHOR]

We introduce a new model of random d-dimensional simplicial complexes, for d ≥ 2 , whose (d - 1) -cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs (The existence of designs; arXiv:1401.3665, 2014), and the proof of the expansion uses techniques developed by Evra and Kaufman in (Bounded degree cosystolic expanders of every dimension; arXiv:1510.00839, 2015). This gives a full solution to a question raised in Dotterrer and Kahle (J Topol Anal 4(4): 499–514, 2012), which was solved in the two-dimensional case by Lubotzky and Meshulam (Adv Math 272: 743–760, 2015). [ABSTRACT FROM AUTHOR]

We prove general results which include classical facts about the 60 lines of Pascal as special cases. Along similar lines we establish analogous results about the configurations of 2,520 conics arising from the mystic octagon. We offer a more combinatorial outlook on these results. Bézout's theorem is the main tool; however, its application is guided by the empirical evidence and computer experiments with the program Cinderella. [ABSTRACT FROM AUTHOR]