The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an n × n square lattice such that no k+1 of them are collinear. The problem was first posed more than 100 years ago for the special case k=2 and has remained open ever since. The general problem was recently resolved in the case k is not small compared to n, as Kovács, Nagy and Szabó proved that the upper bound kn can be attained, provided that k>C√(nlogn) for an absolute constant C. In this paper, we show that (1-2k)kn ≤ f_k(n)≤ kn and (1-3k)kn ≤ f_k(n)≤ kn hold for every even k and odd k, respectively, provided that n is large enough. This is asymptotically tight as k→∞. Previously, only f_k(n)=Ω(kn) was known due to Lefmann. We present further improvements on the lower bounds for constant values of k when k<23 holds. All these bounds are based on randomised algebraic constructions.
For any uniformity r and residue k modulo r, we give an exact characterization of the r-uniform hypergraphs that homomorphically avoid tight cycles of length k modulo r, in terms of colorings of (r-1)-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kamčev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form 𝒞_k^(r)={r-uniform tight cycles of length k modulo r}. We also outline a general strategy to prove that, if 𝒞 is a family of tight-cycle-like hypergraphs (including but not limited to the families 𝒞_k^(r)) for which the above characterization applies, then all sufficiently long C∈𝒞 will have the same Turán density. We demonstrate an application of this framework, proving that there exists an integer L_0 such that for every L>L_0 not divisible by 4, the tight cycle C^(4)_L has Turán density 1/2.
We show that every n-vertex planar graph is contained in the graph obtained from a fan by blowing up each vertex by a complete graph of order O(√(n log² n)). Equivalently, every n-vertex planar graph G has a set X of O(√(n log² n)) vertices such that G−X has bandwidth O(√(n log² n)). We in fact prove the same result for any proper minor-closed class, and we prove more general results that explore the trade-off between X and the bandwidth of G−X. The proofs use three key ingredients. The first is a new local sparsification lemma, which shows that every n-vertex planar graph G has a set of O((n log n)/δ) vertices whose removal results in a graph with local density at most δ. The second is a generalization of a method of Feige and Rao that relates bandwidth and local density using volume-preserving Euclidean embeddings. The third ingredient is graph products, which are a key tool in the extension to any proper minor-closed class.
We show that every graph with twin-width t has chromatic number O(ω k t ) for some integer k t , where ω denotes the clique number.This extends a quasi-polynomial bound from Pilipczuk and Sokołowski and generalizes a result for bounded clique-width graphs by Bonamy and Pilipczuk.The proof uses the main ideas of the quasi-polynomial approach, with a different treatment of the decomposition tree.In particular, we identify two types of extensions of a class of graphs: the delayed-extension (which preserves polynomial χ-boundedness) and the right-extension (which preserves polynomial χ-boundedness under bounded twin-width condition).Our main result is that every bounded twin-width graph is a delayed extension of simpler classes of graphs, each expressed as a bounded union of right extensions of lower twin-width graphs.
A highly influential result of Nikiforov states that if an n-vertex graph G contains at least γn^h copies of a fixed h-vertex graph H, then G contains a blowup of H of order Ω_γ,H(log n). While the dependence on n is optimal, the correct dependence on γ is unknown; all known proofs yield bounds that are polynomial in γ, but the best known upper bound, coming from random graphs, is only logarithmic in γ. It is a major open problem to narrow this gap. We prove that if H is triangle-free, then the logarithmic behavior of the upper bound is the truth. That is, under the assumptions above, G contains a blowup of H of order Ω_H (log n/log(1/γ)). This is the first non-trivial instance where the optimal dependence in Nikiforov's theorem is known. As a consequence, we also prove an upper bound on multicolor Ramsey numbers of blowups of triangle-free graphs, proving that the dependence on the number of colors is polynomial once the blowup is sufficiently large. This shows that, from the perspective of multicolor Ramsey numbers, blowups of fixed triangle-free graphs behave like bipartite graphs.
We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or obtained from a biclique by turning one side into a triangle. Our construction is explicit, canonical, and has the following applications: we obtain a new theorem characterising all Cayley graphs as either essentially 4-connected, cycles, or complete graphs on at most four vertices, and we provide an automatic proof of Tutte’s wheel theorem.
We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruci\'{n}ski. Namely, for any $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$ pairwise disjoint subsets in $\mathbb Z$ of size $r$ contains a subcollection of size $\lfloor n^{1/(2^r-1)}/2\rfloor$ in which every pair of sets are in the same relative position with respect to the linear ordering on $\mathbb Z$. This improves previous bounds of Dudek-Grytczuk-Ruci\'nski and of Anastos-Jin-Kwan-Sudakov and is sharp up to a factor of $2$. For large $r$, we even obtain such a subcollection of size $\lfloor (1-o(1))\cdot n^{1/(2^r-1)}\rfloor$, which is asymptotically tight (here, the $o(1)$-term tends to zero as $r \to \infty$, regardless of the value of $n$). Furthermore, we prove a multiparameter extension of this result where one wants to find a clique of prescribed size $m_P$ for each relative position pattern $P$. Our bound is sharp for all choices of parameters $m_P$, up to a constant factor depending on $r$ only. This answers questions of Anastos-Jin-Kwan-Sudakov and of Dudek-Grytczuk-Ruci\'nski.
The celebrated Erd\H{o}s-Pósa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about the treewidth of graphs containing no induced subgraph isomorphic to the disjoint union of $c$ cycles? Let us call these graphs $c$-perforated. While $1$-perforated graphs have treewidth one, complete graphs and complete bipartite graphs are examples of $2$-perforated graphs with arbitrarily large treewidth. But there are sparse examples, too: Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek constructed $2$-perforated graphs with arbitrarily large treewidth and no induced subgraph isomorphic to $K_3$ or $K_{3,3}$; we call these graphs occultations. Indeed, it turns out that a mild (and inevitable) adjustment of occultations provides examples of $2$-perforated graphs with arbitrarily large treewidth and arbitrarily large girth, which we refer to as full occultations. Our main result shows that the converse also holds: for every $c\geq 1$, a $c$-perforated graph has large treewidth if and only if it contains, as an induced subgraph, either a large complete graph, or a large complete bipartite graph, or a large full occultation. This distinguishes $c$-perforated graphs, among graph classes purely defined by forbidden induced subgraphs, as the first to admit a grid-type theorem incorporating obstructions other than subdivided walls and their line graphs. More generally, for all $c,o\geq 1$, we establish a full characterization of induced subgraph obstructions to bounded treewidth in graphs containing no induced subgraph isomorphic to the disjoint union of $c$ cycles, each of length at least $o+2$.
We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.
For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P)<\alpha\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, \alpha}(n)$ as $n \to \infty$ by showing the existence of positive constants $\beta = \beta(d, \alpha)$ and $\gamma = \gamma(d)$ such that $\beta n^{\frac{d-1}{d+1}} \le c_{d, \alpha}(n) \le \gamma n^{\frac{d-1}{d+1}}$ for $\alpha\geq 2$.
Lovász (1967) showed that two graphs G and H are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph F, the number of homomorphisms from F to G equals the number of homomorphisms from F to H. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).
We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. This in particular settles three of Babai’s conjectures from the 1980s. The first one states that the order of non-alternating, non-abelian composition factors for automorphism groups of graphs of bounded Hadwiger number is bounded. The second one, the subcontraction conjecture, states that a non-trivial minor-closed graph class represents only finitely many non-abelian simple groups. And the third one states that if the order of such a group does not have small prime factors, then the group is obtained by iterated wreath and direct products from abelian groups. Our proof includes a structural analysis of finite edge-transitive graphs.
A [Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set) is a set of points of an Euclidean space that contains a unit line segment in every direction. In 1919, [Besicovitch](https://en.wikipedia.org/wiki/Abram_Samoilovitch_Besicovitch) constructed a Kakeya set of measure zero for every dimension, and in addition, he constructed sets in the plane with arbitrarily small measure such that a unit segment can rotate full 360 degrees within the set. While Kakeya sets can have measure zero, the famous [Kakeya Conjecture](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture) asserts that every Kakeya set in ${\mathbb R}^n$ has both [Hausdorff dimension](https://en.wikipedia.org/wiki/Hausdorff_dimension) and [Minkowski dimension](https://en.wikipedia.org/wiki/Minkowski_dimension) equal to $n$. The conjecture is open for $n\ge 3$. This paper concerns Kakeya sets in the finite setting. In 1999, Wolff conjectured that every Kakeya set in ${\mathbb F}^n$, i.e., a set containing a line in every direction, has size at least $c_n\cdot\lvert {\mathbb F}\rvert^n$. The conjecture was proven by Dvir with $c_n=1/n!$ in 2008 (an exposition of the proof can be found in [this blog post on Terence Tao's blog](https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/)). Subsequently, Ellenberg, Oberlin and Tao proposed studying Kakeya sets over the rings ${\mathbb Z}/p^k{\mathbb Z}$, and Hickman and Wright over ${\mathbb Z}/N{\mathbb Z}$ for an arbitrary $N$. The paper resolves a conjecture of Hickman and Wright by giving an exponential lower bound on the size of a Kakeya set in this most general setting.
This paper investigates big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages. Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees are still not well understood. We show that if there are only finitely many relations of every arity greater than one, then unrestricted relational structures have finite big Ramsey degrees, and give some evidence that this is tight. This is the first time finiteness of big Ramsey degrees has been established for a random structure in an infinite language. Our results represent an important step towards a better understanding of big Ramsey degrees for structures with relations of arity greater than two.
We say a class C of graphs is clean if for every positive integer t there exists a positive integer w ( t ) such that every graph in C with treewidth more than w ( t ) contains an induced subgraph isomorphic to one of the following: the complete graph K t, the complete bipartite graph K t , t, a subdivision of the ( t × t )-wall or the line graph of a subdivision of the ( t × t )-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all H-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph H ) is clean if and only if H is a forest whose components are subdivided stars. Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest H as above, we show that forbidding certain connected graphs containing H as an induced subgraph (rather than H itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer η , a complete description of unavoidable connected induced subgraphs of a connected graph G containing η vertices from a suitably large given set of vertices in G . This is of independent interest, and will be used in subsequent papers in this series.
We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$ contains either $K_{s,s}$ as a subgraph or contains an induced subdivision of $H$. This improves upon a result of K\"uhn and Osthus from 2004 who proved it for graphs whose average degree is at least triply exponential in $s$ and a recent result of Du, Gir\~{a}o, Hunter, McCarty and Scott for graphs with average degree at least singly exponential in $s$. As an application, we prove that the class of graphs that do not contain an induced subdivision of $K_{s,t}$ is polynomially $\chi$-bounded. In the case of $K_{2,3}$, this is the class of theta-free graphs, and answers a question of Davies. Along the way, we also answer a recent question of McCarty, by showing that if $\mathcal{G}$ is a hereditary class of graphs for which there is a polynomial $p$ such that every bipartite $K_{s,s}$-free graph in $\mathcal{G}$ has average degree at most $p(s)$, then more generally, there is a polynomial $p'$ such that every $K_{s,s}$-free graph in $\mathcal{G}$ has average degree at most $p'(s)$. Our main new tool is an induced variant of the K\H{o}v\'ari-S\'os-Tur\'an theorem, which we find to be of independent interest.
We prove a precise min-max theorem for the following problem. Let $G$ be an Eulerian graph with a specified set of edges $S \subseteq E(G)$, and let $b$ be a vertex of $G$. Then what is the maximum integer $k$ so that the edge-set of $G$ can be partitioned into $k$ non-zero $b$-trails? That is, each trail must begin and end at $b$ and contain an odd number of edges from $S$. This theorem is motivated by a connection to vertex-minors and yields two conjectures of Má\v{c}ajová and \v{S}koviera as corollaries.
In a ground-breaking paper solving a conjecture of Erd\H{o}s on the number of $n$-vertex graphs not containing a given even cycle, Morris and Saxton \cite{MS} made a broad conjecture on so-called balanced supersaturation property of a bipartite graph $H$. Ferber, McKinley, and Samotij \cite{FMS} established a weaker version of this conjecture and applied it to derive far-reaching results on the enumeration problem of $H$-free graphs. In this paper, we show that Morris and Saxton's conjecture holds under a very mild assumption about $H$, which is widely believed to hold whenever $H$ contains a cycle. We then use our theorem to obtain enumeration results and general upper bounds on the Tur\'an number of a bipartite $H$ in the random graph $G(n,p)$, the latter being first of its kind.
We present several results in extremal graph and hypergraph theory of topological nature. First, we show that if α>0 and ℓ=Ω(1/αlog1/α) is an odd integer, then every graph G with n vertices and at least n^1+α edges contains an ℓ-subdivision of the complete graph K_t, where t=n^Θ(α). Also, this remains true if in addition the edges of G are properly colored, and one wants to find a rainbow copy of such a subdivision. In the sparser regime, we show that properly edge colored graphs on n vertices with average degree (log n)^2+o(1) contain rainbow cycles, while average degree (log n)^6+o(1) guarantees rainbow subdivisions of K_t for any fixed t, thus improving recent results of Janzer and Jiang et al., respectively. Furthermore, we consider certain topological notions of cycles in pure simplicial complexes (uniform hypergraphs). We show that if G is a 2-dimensional pure simplicial complex (3-graph) with n 1-dimensional and at least n^1+α 2-dimensional faces, then G contains a triangulation of the cylinder and the Möbius strip with O(1/αlog1/α) vertices. We present generalizations of this for higher dimensional pure simplicial complexes as well. In order to prove these results, we consider certain (properly edge colored) graphs and hypergraphs G with strong expansion. We argue that if one randomly samples the vertices (and colors) of G with not too small probability, then many pairs of vertices are connected by a short path whose vertices (and colors) are from the sampled set, with high probability.
