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.
