@inproceedings{14888, abstract = {A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a probabilistic FPT-approach in the number of popular faces.}, author = {De Nooijer, Phoebe and Terziadis, Soeren and Weinberger, Alexandra and Masárová, Zuzana and Mchedlidze, Tamara and Löffler, Maarten and Rote, Günter}, booktitle = {31st International Symposium on Graph Drawing and Network Visualization}, isbn = {9783031492747}, issn = {1611-3349}, location = {Isola delle Femmine, Palermo, Italy}, pages = {18--33}, publisher = {Springer Nature}, title = {{Removing popular faces in curve arrangements}}, doi = {10.1007/978-3-031-49275-4_2}, volume = {14466}, year = {2024}, } @article{14445, abstract = {We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ) is at least 2−d Π d−1k=0ηk. As an application, we show that for every sufficiently thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings. We complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove • an upper bound of (d + 1)/2d on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite d-dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and • an upper bound of (d + 1)/2d + ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical building associated with GLd+2(Fq) for any ε > 0 and sufficiently large q. This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.}, author = {Wagner, Uli and Wild, Pascal}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, number = {2}, pages = {675--717}, publisher = {Springer Nature}, title = {{Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes}}, doi = {10.1007/s11856-023-2521-9}, volume = {256}, year = {2023}, } @article{14660, abstract = {The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set 𝑆⊂ℝ𝑑, then there are at most 2𝑑 points of 𝑆 whose convex hull contains the origin in the interior. Bárány, Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem. Let 𝑄 be a convex polytope in ℝ𝑑 containing the standard Euclidean unit ball 𝐁𝑑. Then there exist at most 2𝑑 vertices of 𝑄 whose convex hull 𝑄′ satisfies 𝑟𝐁𝑑⊂𝑄′ with 𝑟⩾𝑑−2𝑑. They conjectured that 𝑟⩾𝑐𝑑−1∕2 holds with a universal constant 𝑐>0. We prove 𝑟⩾15𝑑2, the first polynomial lower bound on 𝑟. Furthermore, we show that 𝑟 is not greater than 2/√𝑑.}, author = {Ivanov, Grigory and Naszódi, Márton}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, publisher = {London Mathematical Society}, title = {{Quantitative Steinitz theorem: A polynomial bound}}, doi = {10.1112/blms.12965}, year = {2023}, } @article{14737, abstract = {John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^{d}$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called Löwner’s ellipsoid, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the Löwner problems is more intricate than it is in the setting of convex bodies.}, author = {Ivanov, Grigory and Naszódi, Márton}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, keywords = {General Mathematics}, number = {23}, pages = {20613--20669}, publisher = {Oxford University Press}, title = {{Functional John and Löwner conditions for pairs of log-concave functions}}, doi = {10.1093/imrn/rnad210}, volume = {2023}, year = {2023}, } @article{13270, abstract = {Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform δ>0 such that, at any step of the subdivision, all the triangle angles lie in the interval (δ,π−δ) . Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.}, author = {Brunck, Florestan R}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, number = {3}, pages = {1059--1089}, publisher = {Springer Nature}, title = {{Iterated medial triangle subdivision in surfaces of constant curvature}}, doi = {10.1007/s00454-023-00500-5}, volume = {70}, year = {2023}, } @phdthesis{13331, abstract = {The extension of extremal combinatorics to the setting of exterior algebra is a work in progress that gained attention recently. In this thesis, we study the combinatorial structure of exterior algebra by introducing a dictionary that translates the notions from the set systems into the framework of exterior algebra. We show both generalizations of celebrated Erdös--Ko--Rado theorem and Hilton--Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms. }, author = {Köse, Seyda}, issn = {2791-4585}, pages = {26}, publisher = {Institute of Science and Technology Austria}, title = {{Exterior algebra and combinatorics}}, doi = {10.15479/at:ista:13331}, year = {2023}, } @article{13969, abstract = {Bundling crossings is a strategy which can enhance the readability of graph drawings. In this paper we consider good drawings, i.e., we require that any two edges have at most one common point which can be a common vertex or a crossing. Our main result is that there is a polynomial-time algorithm to compute an 8-approximation of the bundled crossing number of a good drawing with no toothed hole. In general the number of toothed holes has to be added to the 8-approximation. In the special case of circular drawings the approximation factor is 8, this improves upon the 10-approximation of Fink et al. [14]. Our approach also works with the same approximation factor for families of pseudosegments, i.e., curves intersecting at most once. We also show how to compute a 9/2-approximation when the intersection graph of the pseudosegments is bipartite and has no toothed hole.}, author = {Arroyo Guevara, Alan M and Felsner, Stefan}, issn = {1526-1719}, journal = {Journal of Graph Algorithms and Applications}, number = {6}, pages = {433--457}, publisher = {Brown University}, title = {{Approximating the bundled crossing number}}, doi = {10.7155/jgaa.00629}, volume = {27}, year = {2023}, } @article{13974, abstract = {The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r−1)+1 points in Rd, one can find a partition X=X1∪⋯∪Xr of X, such that the convex hulls of the Xi, i=1,…,r, all share a common point. In this paper, we prove a trengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊n/3⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊n/6⌋pairwise crossing triangles. Our result generalizes to a result about simplices in Rd, d≥2.}, author = {Fulek, Radoslav and Gärtner, Bernd and Kupavskii, Andrey and Valtr, Pavel and Wagner, Uli}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, publisher = {Springer Nature}, title = {{The crossing Tverberg theorem}}, doi = {10.1007/s00454-023-00532-x}, year = {2023}, } @article{9651, abstract = {We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞). Two separated nets are called ϕ-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most ϕ(R). We show that the spectrum of ϕ-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation, coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞. We further undertake a comparison of our notion of ϕ-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz equivalence.}, author = {Dymond, Michael and Kaluza, Vojtech}, issn = {1572-9168}, journal = {Geometriae Dedicata}, publisher = {Springer Nature}, title = {{Divergence of separated nets with respect to displacement equivalence}}, doi = {10.1007/s10711-023-00862-3}, year = {2023}, } @article{9652, abstract = {In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate that such notions also give rise to distinct equivalence classes. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions ρ:[0,1]d→(0,∞). In the present work we obtain stronger types of non-realisable densities.}, author = {Dymond, Michael and Kaluza, Vojtech}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, keywords = {Lipschitz, bilipschitz, bounded displacement, modulus of continuity, separated net, non-realisable density, Burago--Kleiner construction}, pages = {501--554}, publisher = {Springer Nature}, title = {{Highly irregular separated nets}}, doi = {10.1007/s11856-022-2448-6}, volume = {253}, year = {2023}, } @article{11999, abstract = {A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ, it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles.}, author = {Arroyo Guevara, Alan M and Klute, Fabian and Parada, Irene and Vogtenhuber, Birgit and Seidel, Raimund and Wiedera, Tilo}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {745–770}, publisher = {Springer Nature}, title = {{Inserting one edge into a simple drawing is hard}}, doi = {10.1007/s00454-022-00394-9}, volume = {69}, year = {2023}, } @article{12563, abstract = {he approximate graph coloring problem, whose complexity is unresolved in most cases, concerns finding a c-coloring of a graph that is promised to be k-colorable, where c≥k. This problem naturally generalizes to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyze the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph coloring and promise graph homomorphism problems.}, author = {Krokhin, Andrei and Opršal, Jakub and Wrochna, Marcin and Živný, Stanislav}, issn = {1095-7111}, journal = {SIAM Journal on Computing}, keywords = {General Mathematics, General Computer Science}, number = {1}, pages = {38--79}, publisher = {Society for Industrial & Applied Mathematics}, title = {{Topology and adjunction in promise constraint satisfaction}}, doi = {10.1137/20m1378223}, volume = {52}, year = {2023}, } @article{12680, abstract = {The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines.}, author = {Ivanov, Grigory and Köse, Seyda}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {6}, publisher = {Elsevier}, title = {{Erdős-Ko-Rado and Hilton-Milner theorems for two-forms}}, doi = {10.1016/j.disc.2023.113363}, volume = {346}, year = {2023}, } @article{12833, abstract = {The input to the token swapping problem is a graph with vertices v1, v2, . . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token swapping on a tree, also known as “sorting with a transposition tree,” is not known to be in P nor NP-complete. We present some partial results: 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a “happy leaf”), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves—as all known approximation algorithms for the problem do—has approximation factor at least 4/3. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem—weighted coloured token swapping—is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.}, author = {Biniaz, Ahmad and Jain, Kshitij and Lubiw, Anna and Masárová, Zuzana and Miltzow, Tillmann and Mondal, Debajyoti and Naredla, Anurag Murty and Tkadlec, Josef and Turcotte, Alexi}, issn = {1365-8050}, journal = {Discrete Mathematics and Theoretical Computer Science}, number = {2}, publisher = {EPI Sciences}, title = {{Token swapping on trees}}, doi = {10.46298/DMTCS.8383}, volume = {24}, year = {2023}, } @article{10776, abstract = {Let K be a convex body in Rn (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K∩h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p0 is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n≥2, there are always at least three distinct barycentric cuts through the point p0∈K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p0 are guaranteed if n≥3.}, author = {Patakova, Zuzana and Tancer, Martin and Wagner, Uli}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {1133--1154}, publisher = {Springer Nature}, title = {{Barycentric cuts through a convex body}}, doi = {10.1007/s00454-021-00364-7}, volume = {68}, year = {2022}, } @article{10887, abstract = {We introduce a new way of representing logarithmically concave functions on Rd. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every s>0, we define a class of non-negative functions on Rd derived from ellipsoids in Rd+1. For any log-concave function f on Rd , and any fixed s>0, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to f, and we call it the John s-function of f. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John s-functions converge to characteristic functions of ellipsoids as s tends to zero and to Gaussian densities as s tends to infinity. As an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant cd multiple of the integral of the pointwise minimum of a properly chosen subfamily of size 3d+2, where cd depends only on d.}, author = {Ivanov, Grigory and Naszódi, Márton}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {11}, publisher = {Elsevier}, title = {{Functional John ellipsoids}}, doi = {10.1016/j.jfa.2022.109441}, volume = {282}, year = {2022}, } @inproceedings{11185, abstract = {Bundling crossings is a strategy which can enhance the readability of graph drawings. In this paper we consider bundlings for families of pseudosegments, i.e., simple curves such that any two have share at most one point at which they cross. Our main result is that there is a polynomial-time algorithm to compute an 8-approximation of the bundled crossing number of such instances (up to adding a term depending on the facial structure). This 8-approximation also holds for bundlings of good drawings of graphs. In the special case of circular drawings the approximation factor is 8 (no extra term), this improves upon the 10-approximation of Fink et al. [6]. We also show how to compute a 92-approximation when the intersection graph of the pseudosegments is bipartite.}, author = {Arroyo Guevara, Alan M and Felsner, Stefan}, booktitle = {WALCOM 2022: Algorithms and Computation}, isbn = {9783030967307}, issn = {1611-3349}, location = {Jember, Indonesia}, pages = {383--395}, publisher = {Springer Nature}, title = {{Approximating the bundled crossing number}}, doi = {10.1007/978-3-030-96731-4_31}, volume = {13174}, year = {2022}, } @article{11435, abstract = {We introduce a new variant of quantitative Helly-type theorems: the minimal homothetic distance of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the diameter. If $K$ is the intersection of finitely many convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d)^3{diam}(K)$. The best previously known estimate, due to Brazitikos [Bull. Hellenic Math. Soc., 62 (2018), pp. 19--25], is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$ conjectured by Bárány, Katchalski, and Pach [Proc. Amer. Math. Soc., 86 (1982), pp. 109--114] cannot be improved. The bounds above follow from our key result that concerns sparse approximation of a convex polytope by the convex hull of a well-chosen subset of its vertices: Assume that $Q \subset {\mathbb R}^d$ is a polytope whose centroid is the origin. Then there exist at most 2d vertices of $Q$ whose convex hull $Q^{\prime \prime}$ satisfies $Q \subset - 8d^3 Q^{\prime \prime}.$}, author = {Ivanov, Grigory and Naszodi, Marton}, issn = {0895-4801}, journal = {SIAM Journal on Discrete Mathematics}, number = {2}, pages = {951--957}, publisher = {Society for Industrial and Applied Mathematics}, title = {{A quantitative Helly-type theorem: Containment in a homothet}}, doi = {10.1137/21M1403308}, volume = {36}, year = {2022}, } @article{11593, abstract = {A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z2 -genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t×t grid or one of the following so-called t -Kuratowski graphs: K3,t, or t copies of K5 or K3,3 sharing at most two common vertices. We show that the Z2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its Z2-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani–Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler Z2-genus of graphs.}, author = {Fulek, Radoslav and Kynčl, Jan}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {425--447}, publisher = {Springer Nature}, title = {{The Z2-Genus of Kuratowski minors}}, doi = {10.1007/s00454-022-00412-w}, volume = {68}, year = {2022}, } @phdthesis{11777, abstract = {In this dissertation we study coboundary expansion of simplicial complex with a view of giving geometric applications. Our main novel tool is an equivariant version of Gromov's celebrated Topological Overlap Theorem. The equivariant topological overlap theorem leads to various geometric applications including a quantitative non-embeddability result for sufficiently thick buildings (which partially resolves a conjecture of Tancer and Vorwerk) and an improved lower bound on the pair-crossing number of (bounded degree) expander graphs. Additionally, we will give new proofs for several known lower bounds for geometric problems such as the number of Tverberg partitions or the crossing number of complete bipartite graphs. For the aforementioned applications one is naturally lead to study expansion properties of joins of simplicial complexes. In the presence of a special certificate for expansion (as it is the case, e.g., for spherical buildings), the join of two expanders is an expander. On the flip-side, we report quite some evidence that coboundary expansion exhibits very non-product-like behaviour under taking joins. For instance, we exhibit infinite families of graphs $(G_n)_{n\in \mathbb{N}}$ and $(H_n)_{n\in\mathbb{N}}$ whose join $G_n*H_n$ has expansion of lower order than the product of the expansion constant of the graphs. Moreover, we show an upper bound of $(d+1)/2^d$ on the normalized coboundary expansion constants for the complete multipartite complex $[n]^{*(d+1)}$ (under a mild divisibility condition on $n$). Via the probabilistic method the latter result extends to an upper bound of $(d+1)/2^d+\varepsilon$ on the coboundary expansion constant of the spherical building associated with $\mathrm{PGL}_{d+2}(\mathbb{F}_q)$ for any $\varepsilon>0$ and sufficiently large $q=q(\varepsilon)$. This disproves a conjecture of Lubotzky, Meshulam and Mozes -- in a rather strong sense. By improving on existing lower bounds we make further progress towards closing the gap between the known lower and upper bounds on the coboundary expansion constants of $[n]^{*(d+1)}$. The best improvements we achieve using computer-aided proofs and flag algebras. The exact value even for the complete $3$-partite $2$-dimensional complex $[n]^{*3}$ remains unknown but we are happy to conjecture a precise value for every $n$. %Moreover, we show that a previously shown lower bound on the expansion constant of the spherical building associated with $\mathrm{PGL}_{2}(\mathbb{F}_q)$ is not tight. In a loosely structured, last chapter of this thesis we collect further smaller observations related to expansion. We point out a link between discrete Morse theory and a technique for showing coboundary expansion, elaborate a bit on the hardness of computing coboundary expansion constants, propose a new criterion for coboundary expansion (in a very dense setting) and give one way of making the folklore result that expansion of links is a necessary condition for a simplicial complex to be an expander precise.}, author = {Wild, Pascal}, isbn = {978-3-99078-021-3}, issn = {2663-337X}, pages = {170}, publisher = {Institute of Science and Technology}, title = {{High-dimensional expansion and crossing numbers of simplicial complexes}}, doi = {10.15479/at:ista:11777}, year = {2022}, }