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Turán Density of Long Tight Cycle Minus One Hyperedge Combinatorica (IF 1.1) Pub Date : 2024-04-17 József Balogh, Haoran Luo
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Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods Combinatorica (IF 1.1) Pub Date : 2024-04-17 Felix Klingelhoefer, Alantha Newman
The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of
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Solution to a Problem of Grünbaum on the Edge Density of 4-Critical Planar Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-17 Zdeněk Dvořák, Carl Feghali
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Small Subgraphs with Large Average Degree Combinatorica (IF 1.1) Pub Date : 2024-04-15 Oliver Janzer, Benny Sudakov, István Tomon
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A Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-15 Alexandr Kostochka, Ruth Luo, Grace McCourt
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Globally Linked Pairs of Vertices in Generic Frameworks Combinatorica (IF 1.1) Pub Date : 2024-04-08 Tibor Jordán, Soma Villányi
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Induced Subgraphs and Tree Decompositions VIII: Excluding a Forest in (Theta, Prism)-Free Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-08 Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl
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Bounded-Diameter Tree-Decompositions Combinatorica (IF 1.1) Pub Date : 2024-04-08 Eli Berger, Paul Seymour
When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to
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The Number of Topological Types of Trees Combinatorica (IF 1.1) Pub Date : 2024-04-04
Abstract Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly \(\aleph _1\) distinct topological types of countable trees. In general, for any infinite cardinal \(\kappa \) there are exactly \(\kappa ^+\) distinct topological types of trees of size \(\kappa \) . This solves a problem of van der Holst from 2005.
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Lattice Path Matroids and Quotients Combinatorica (IF 1.1) Pub Date : 2024-04-04 Carolina Benedetti-Velásquez, Kolja Knauer
We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative
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Flashes and Rainbows in Tournaments Combinatorica (IF 1.1) Pub Date : 2024-04-04 António Girão, Freddie Illingworth, Lukas Michel, Michael Savery, Alex Scott
Colour the edges of the complete graph with vertex set \({\{1, 2, \dotsc , n\}}\) with an arbitrary number of colours. What is the smallest integer f(l, k) such that if \(n > f(l,k)\) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that \(f(l, k) = l^{k - 1}\) and proved this for \(l \ge (3 k)^{2 k}\)
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Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree Combinatorica (IF 1.1) Pub Date : 2024-04-04 Shirshendu Ganguly, Ella Hiesmayr, Kyeongsik Nam
Let N be the number of triangles in an Erdős–Rényi graph \({\mathcal {G}}(n,p)\) on n vertices with edge density \(p=d/n,\) where \(d>0\) is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean \({d^3}/{6}\) as \(n\rightarrow \infty \). We address the upper tail problem for N, namely, we investigate how fast k must grow, so that \({\mathbb {P}}(N\ge k)\)
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Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-04
Abstract It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a
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Boolean Function Analysis on High-Dimensional Expanders Combinatorica (IF 1.1) Pub Date : 2024-03-18 Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha
We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition
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Ramsey Problems for Monotone Paths in Graphs and Hypergraphs Combinatorica (IF 1.1) Pub Date : 2024-02-28 Lior Gishboliner, Zhihan Jin, Benny Sudakov
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The Ungar Games Combinatorica (IF 1.1) Pub Date : 2024-02-21 Colin Defant, Noah Kravitz, Nathan Williams
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An Upper Bound for the Height of a Tree with a Given Eigenvalue Combinatorica (IF 1.1) Pub Date : 2024-02-02 Artūras Dubickas
In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest
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On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries Combinatorica (IF 1.1) Pub Date : 2024-01-05
Abstract Given a (thick) irreducible spherical building \(\Omega \) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long
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Euler’s Theorem for Regular CW-Complexes Combinatorica (IF 1.1) Pub Date : 2024-01-05 Richard H. Hammack, Paul C. Kainen
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A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces Combinatorica (IF 1.1) Pub Date : 2023-12-19 Vuong Bui, Hamid Reza Daneshpajouh
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Tight Bound on Treedepth in Terms of Pathwidth and Longest Path Combinatorica (IF 1.1) Pub Date : 2023-12-19
Abstract We show that every graph with pathwidth strictly less than a that contains no path on \(2^b\) vertices as a subgraph has treedepth at most 10ab. The bound is best possible up to a constant factor.
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Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs Combinatorica (IF 1.1) Pub Date : 2023-12-19 Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf
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A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution Combinatorica (IF 1.1) Pub Date : 2023-11-27 Nguyen Cong Minh, Thanh Vu
Let I(G) be the edge ideal of a simple graph G. We prove that \(I(G)^2\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 3\). Similarly, we show that \(I(G)^3\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 4\). We deduce these characterizations by establishing a general formula for the regularity of powers
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A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups Combinatorica (IF 1.1) Pub Date : 2023-11-27 Tao Zhang
Fuglede’s conjecture states that a subset \(\Omega \subseteq \mathbb {R}^{n}\) with positive and finite Lebesgue measure is a spectral set if and only if it tiles \(\mathbb {R}^{n}\) by translation. However, this conjecture does not hold in both directions for \(\mathbb {R}^n\), \(n\ge 3\). While the conjecture remains unsolved in \(\mathbb {R}\) and \(\mathbb {R}^2\), cyclic groups are instrumental
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Universal Planar Graphs for the Topological Minor Relation Combinatorica (IF 1.1) Pub Date : 2023-11-21 Florian Lehner
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Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs Combinatorica (IF 1.1) Pub Date : 2023-11-21 Jan Corsten, Walner Mendonça
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Kempe Equivalent List Colorings Combinatorica (IF 1.1) Pub Date : 2023-11-16 Daniel W. Cranston, Reem Mahmoud
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A Book Proof of the Middle Levels Theorem Combinatorica (IF 1.1) Pub Date : 2023-11-06 Torsten Mütze
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On Unique Sums in Abelian Groups Combinatorica (IF 1.1) Pub Date : 2023-11-01 Benjamin Bedert
Let A be a subset of the cyclic group \({\textbf{Z}}/p{\textbf{Z}}\) with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in \(A+A\), meaning that for every two elements \(a_1,a_2\in A\), there exist \(a_1',a_2'\in A\) such that \(a_1+a_2=a_1'+a_2'\) and \(\{a_1,a_2\}\ne \{a_1',a_2'\}\). Let m(p) be the size of a smallest subset of \({\textbf{Z}}/p{\textbf{Z}}\)
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Sweeps, Polytopes, Oriented Matroids, and Allowable Graphs of Permutations Combinatorica (IF 1.1) Pub Date : 2023-10-23 Arnau Padrol, Eva Philippe
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A Structural Theorem for Sets with Few Triangles Combinatorica (IF 1.1) Pub Date : 2023-10-12 Sam Mansfield, Jonathan Passant
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On the Maximum of the Sum of the Sizes of Non-trivial Cross-Intersecting Families Combinatorica (IF 1.1) Pub Date : 2023-10-12 P. Frankl
Let \(n \ge 2k \ge 4\) be integers, \({[n]\atopwithdelims ()k}\) the collection of k-subsets of \([n] = \{1, \ldots , n\}\). Two families \({\mathcal {F}}, {\mathcal {G}} \subset {[n]\atopwithdelims ()k}\) are said to be cross-intersecting if \(F \cap G \ne \emptyset \) for all \(F \in {\mathcal {F}}\) and \(G \in {\mathcal {G}}\). A family is called non-trivial if the intersection of all its members
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A Solution to Babai’s Problems on Digraphs with Non-diagonalizable Adjacency Matrix Combinatorica (IF 1.1) Pub Date : 2023-09-29 Yuxuan Li, Binzhou Xia, Sanming Zhou, Wenying Zhu
The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable
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Value Distributions of Perfect Nonlinear Functions Combinatorica (IF 1.1) Pub Date : 2023-09-29 Lukas Kölsch, Alexandr Polujan
In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case
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A Generalization of the Chevalley–Warning and Ax–Katz Theorems with a View Towards Combinatorial Number Theory Combinatorica (IF 1.1) Pub Date : 2023-09-29 David J. Grynkiewicz
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Induced Subgraphs of Induced Subgraphs of Large Chromatic Number Combinatorica (IF 1.1) Pub Date : 2023-09-25 António Girão, Freddie Illingworth, Emil Powierski, Michael Savery, Alex Scott, Youri Tamitegama, Jane Tan
We prove that, for every graph F with at least one edge, there is a constant \(c_F\) such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F-free induced subgraph has chromatic number at most \(c_F\). This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every \(r\geqslant
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Effective Results on the Size and Structure of Sumsets Combinatorica (IF 1.1) Pub Date : 2023-09-18 Andrew Granville, George Shakan, Aled Walker
Let \(A \subset {\mathbb {Z}}^d\) be a finite set. It is known that NA has a particular size (\(\vert NA\vert = P_A(N)\) for some \(P_A(X) \in {\mathbb {Q}}[X]\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results
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Integer Multiflows in Acyclic Planar Digraphs Combinatorica (IF 1.1) Pub Date : 2023-09-19 Guyslain Naves
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Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path Combinatorica (IF 1.1) Pub Date : 2023-09-15 Alex Scott, Paul Seymour, Sophie Spirkl
A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a \(P_5\)-free graph with clique number \(\omega \ge 3\) has chromatic number at most \(\omega ^{\log _2(\omega )}\). The best previous result was an exponential upper bound \((5/27)3^{\omega }\), due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds
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Prague Dimension of Random Graphs Combinatorica (IF 1.1) Pub Date : 2023-09-06 He Guo, Kalen Patton, Lutz Warnke
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The Strong Nine Dragon Tree Conjecture is True for $$d \le k + 1$$ Combinatorica (IF 1.1) Pub Date : 2023-08-21 Sebastian Mies, Benjamin Moore
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On Bounded Degree Graphs with Large Size-Ramsey Numbers Combinatorica (IF 1.1) Pub Date : 2023-08-21 Konstantin Tikhomirov
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A Characterization of Edge-Ordered Graphs with Almost Linear Extremal Functions Combinatorica (IF 1.1) Pub Date : 2023-08-18 Gaurav Kucheriya, Gábor Tardos
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Separating Polynomial $$\chi $$ -Boundedness from $$\chi $$ -Boundedness Combinatorica (IF 1.1) Pub Date : 2023-08-09 Marcin Briański, James Davies, Bartosz Walczak
Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function \(f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}\) with \(f(1)=1\) and \(f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right) \), we construct a hereditary class of graphs \({\mathcal {G}}\) such that the maximum chromatic number of a graph in \({\mathcal {G}}\) with clique number
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Stability Through Non-Shadows Combinatorica (IF 1.1) Pub Date : 2023-08-04 Jun Gao, Hong Liu, Zixiang Xu
We study families \({\mathcal {F}}\subseteq 2^{[n]}\) with restricted intersections and prove a conjecture of Snevily in a stronger form for large n. We also obtain stability results for Kleitman’s isodiametric inequality and families with bounded set-wise differences. Our proofs introduce a new twist to the classical linear algebra method, harnessing the non-shadows of \({\mathcal {F}}\), which may
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Weak Saturation of Multipartite Hypergraphs Combinatorica (IF 1.1) Pub Date : 2023-07-27 Denys Bulavka, Martin Tancer, Mykhaylo Tyomkyn
Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weakly H-saturated in F if the edges in \(E(F)\setminus E(G)\) admit an ordering \(e_1,\ldots , e_k\) so that for all \(i\in [k]\) the hypergraph \(G\cup \{e_1,\ldots ,e_i\}\) contains an isomorphic copy of H which in turn contains the edge \(e_i\). The weak saturation number of H in F is the smallest
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A Necessary and Sufficient Condition for $$(2d-2)$$ -Transversals in $$\mathbb {R}^{2d}$$ Combinatorica (IF 1.1) Pub Date : 2023-07-27 Daniel McGinnis
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A Factor Matching of Optimal Tail Between Poisson Processes Combinatorica (IF 1.1) Pub Date : 2023-07-25 Ádám Timár
Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., a measurable function of the point configurations that commutes with translations), and with the property that the distance between two matched configuration points has a tail distribution that decays
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Spherical Two-Distance Sets and Eigenvalues of Signed Graphs Combinatorica (IF 1.1) Pub Date : 2023-07-21 Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao
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Pure Pairs. V. Excluding Some Long Subdivision Combinatorica (IF 1.1) Pub Date : 2023-06-16 Alex Scott, Paul Seymour, Sophie Spirkl
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A combinatorial proof of a sumset conjecture of Furstenberg Combinatorica (IF 1.1) Pub Date : 2023-06-14 Daniel Glasscock, Joel Moreira, Florian K. Richter
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if \(\log r/\log s\) is irrational and X and Y are \(\times r\) - and \(\times s\)-invariant subsets of [0, 1], respectively, then \(\dim _{\text {H}}(X + Y ) = \min (1, \dim _{\text {H}}X + \dim _{\text {H}}Y )\). Our main result yields information on the size of the sumset \(\lambda X
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Maximal 3-Wise Intersecting Families Combinatorica (IF 1.1) Pub Date : 2023-06-13 József Balogh, Ce Chen, Kevin Hendrey, Ben Lund, Haoran Luo, Casey Tompkins, Tuan Tran
A family \({\mathcal {F}}\) on ground set \([n]:=\{1,2,\ldots , n\}\) is maximal k-wise intersecting if every collection of at most k sets in \({\mathcal {F}}\) has non-empty intersection, and no other set can be added to \({\mathcal {F}}\) while maintaining this property. In 1974, Erdős and Kleitman asked for the minimum size of a maximal k-wise intersecting family. We answer their question for \(k=3\)
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The Asymptotic Number of Score Sequences Combinatorica (IF 1.1) Pub Date : 2023-06-13 Brett Kolesnik
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A Large Family of Maximum Scattered Linear Sets of $${{\,\mathrm{{PG}}\,}}(1,q^n)$$ and Their Associated MRD Codes Combinatorica (IF 1.1) Pub Date : 2023-06-13 G. Longobardi, Giuseppe Marino, Rocco Trombetti, Yue Zhou
Linear sets in projective spaces over finite fields were introduced by Lunardon (Geom Dedic 75(3):245–261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric codes, etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line
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Enclosing Depth and Other Depth Measures Combinatorica (IF 1.1) Pub Date : 2023-06-13 Patrick Schnider
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A Recursive Theta Body for Hypergraphs Combinatorica (IF 1.1) Pub Date : 2023-06-13 Davi Castro-Silva, Fernando Mário de Oliveira Filho, Lucas Slot, Frank Vallentin
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An Improved Bound for the Linear Arboricity Conjecture Combinatorica (IF 1.1) Pub Date : 2023-06-13 Richard Lang, Luke Postle
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$$\Gamma $$ -Graphic Delta-Matroids and Their Applications Combinatorica (IF 1.1) Pub Date : 2023-06-13 Donggyu Kim, Duksang Lee, Sang-il Oum
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A Unifying Framework for the $$\nu $$ -Tamari Lattice and Principal Order Ideals in Young’s Lattice Combinatorica (IF 1.1) Pub Date : 2023-06-13 Matias von Bell, Rafael S. González D’León, Francisco A. Mayorga Cetina, Martha Yip