Sharper Bounds and Structural Results for Minimally Nonlinear 0-1 Matrices

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally nonlinear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the number of ones and the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones.

On the Existence of an Extremal Function in the Delsarte Extremal Problem

This paper is concerned with a Delsarte-type extremal problem. Denote by $${\mathcal {P}}(G)$$ P ( G ) the set of positive definite continuous functions on a locally compact abelian group G. We consider the function class, which was originally introduced by Gorbachev, $$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$ G ( W , Q ) G = f ∈ P ( G ) ∩ L 1 ( G ) : f ( 0 ) = 1 , supp f + ⊆ W , supp f ^ ⊆ Q where $$W\subseteq G$$ W ⊆ G is closed and of finite Haar measure and $$Q\subseteq {\widehat{G}}$$ Q ⊆ G ^ is compact. We also consider the related Delsarte-type problem of finding the extremal quantity $$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$ D ( W , Q ) G = sup ∫ G f ( g ) d λ G ( g ) : f ∈ G ( W , Q ) G . The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem $${\mathcal {D}}(W,Q)_G$$ D ( W , Q ) G . The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $$G={\mathbb {R}}^d$$ G = R d . So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

Sharp bounds for decomposing graphs into edges and triangles

For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

Extremal problems for convex geometric hypergraphs and ordered hypergraphs

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

Forbidden formations in 0-1 matrices

Keszegh (2009) proved that the extremal function $ex(n, P)$ of any forbidden light $2$-dimensional 0-1 matrix $P$ is at most quasilinear in $n$, using a reduction to generalized Davenport-Schinzel sequences. We extend this result to multidimensional matrices by proving that any light $d$-dimensional 0-1 matrix $P$ has extremal function $ex(n, P,d) = O(n^{d-1}2^{\alpha(n)^{t}})$ for some constant $t$ that depends on $P$. To prove this result, we introduce a new family of patterns called $(P, s)$-formations, which are a generalization of $(r, s)$-formations, and we prove upper bounds on their extremal functions. In many cases, including permutation matrices $P$ with at least two ones, we are able to show that our $(P, s)$-formation upper bounds are tight.