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1. On Single-Distance Graphs on the Rational Points in Euclidean Spaces

   https://doi.org/10.4153/S0008439520000181  

   Bau, Sheng; Johnson, Peter; Noble, Matt (March 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract For positive integers n and d > 0, let $$G(\mathbb {Q}^n,\; d)$$ denote the graph whose vertices are the set of rational points $$\mathbb {Q}^n$$ , with $$u,v \in \mathbb {Q}^n$$ being adjacent if and only if the Euclidean distance between u and v is equal to d . Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $$\mathbb {Q}^n$$ . In this paper, we show that a space $$\mathbb {Q}^n$$ has the property that all pairs of non-trivial distance graphs $$G(\mathbb {Q}^n,\; d_1)$$ and $$G(\mathbb {Q}^n,\; d_2)$$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $$G(\mathbb {Q}^n,\; d)$$ . 

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2. Almost Sharp Bounds on the Number of Discrete Chains in the Plane

   Frankl, N; Kupavskii, A. (June 2020, Proceedings of 36th International Symposium on Computational Geometry (SoCG 2020))

   The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ=(δ₁,… ,δ_k) of k distances, a (k+1)-tuple (p₁,… ,p_{k+1}) of distinct points in ℝ^d is called a (k,δ)-chain if ‖p_j-p_{j+1}‖ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3. 

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3. A completion of the proof of the Edge-statistics Conjecture

   https://doi.org/10.19086/aic.12047  

   Fox, Jacob; Sauermann, Lisa (February 2020, Advances in Combinatorics)

   Extremal combinatorics often deals with problems of maximizing a specific quantity related to substructures in large discrete structures. The first question of this kind that comes to one's mind is perhaps determining the maximum possible number of induced subgraphs isomorphic to a fixed graph $$H$$ in an $$n$$-vertex graph. The asymptotic behavior of this number is captured by the limit of the ratio of the maximum number of induced subgraphs isomorphic to $$H$$ and the number of all subgraphs with the same number vertices as $$H$$; this quantity is known as the _inducibility_ of $$H$$. More generally, one can define the inducibility of a family of graphs in the analogous way.Among all graphs with $$k$$ vertices, the only two graphs with inducibility equal to one are the empty graph and the complete graph. However, how large can the inducibility of other graphs with $$k$$ vertices be? Fix $$k$$, consider a graph with $$n$$ vertices join each pair of vertices independently by an edge with probability $$\binom{k}{2}^{-1}$$. The expected number of $$k$$-vertex induced subgraphs with exactly one edge is $$e^{-1}+o(1)$$. So, the inducibility of large graphs with a single edge is at least $$e^{-1}+o(1)$$. This article establishes that this bound is the best possible in the following stronger form, which proves a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn: the inducibility of the family of $$k$$-vertex graphs with exactly $$l$$ edges where $$0<\binom{k}{2}$$ is at most $$e^{-1}+o(1)$$. The example above shows that this is tight for $l=1$ and it can be also shown to be tight for $l=k-1$. The conjecture was known to be true in the regime where $$l$$ is superlinearly bounded away from $$0$$ and $$\binom{k}{2}$$, for which the sum of the inducibilities goes to zero, and also in the regime where $$l$$ is bounded away from $$0$$ and $$\binom{k}{2}$$ by a sufficiently large linear function. The article resolves the hardest cases where $$l$$ is linearly close to $$0$$ or close to $$\binom{k}{2}$$, and provides generalizations to hypergraphs. 

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4. Zarankiewicz’s problem for semilinear hypergraphs

   https://doi.org/10.1017/fms.2021.52  

   Basit, Abdul; Chernikov, Artem; Starchenko, Sergei; Tao, Terence; Tran, Chieu-Minh (January 2021, Forum of Mathematics, Sigma)

   Abstract A bipartite graph $$H = \left (V_1, V_2; E \right )$$ with $$\lvert V_1\rvert + \lvert V_2\rvert = n$$ is semilinear if $$V_i \subseteq \mathbb {R}^{d_i}$$ for some $$d_i$$ and the edge relation E consists of the pairs of points $$(x_1, x_2) \in V_1 \times V_2$$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $$d_1 + d_2$$ variables for some s . We show that for a fixed k , the number of edges in a $$K_{k,k}$$ -free semilinear H is almost linear in n , namely $$\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ for any $$\varepsilon> 0$$ ; and more generally, $$\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$$ for a $$K_{k, \dotsc ,k}$$ -free semilinear r -partite r -uniform hypergraph. As an application, we obtain the following incidence bound: given $$n_1$$ points and $$n_2$$ open boxes with axis-parallel sides in $$\mathbb {R}^d$$ such that their incidence graph is $$K_{k,k}$$ -free, there can be at most $$O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o -minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner). 

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5. Online Spanners in Metric Spaces

   Bhore, Sujoy; Filtser, Arnold; Khodabandeh, Hadi; Toth, Csaba D. (September 2022, 30th Annual European Symposium on Algorithms (ESA 2022))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

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