Lengths of extremal square-free ternary words

A square-free word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words

Generating square-free words efficiently

We study a simple algorithm generating square-free words from a random source. The source produces uniformly distributed random letters from a k-ary alphabet, and the algorithm outputs a (k+1)-ary square-free word. We are interested in the "conversion ratio" between the lengths of the input random word and the output square-free word. For any k≥3 we prove the expected value of this ratio to be a constant and calculate it up to an O(1/k5) term. For the extremal case of ternary square-free words, we suggest this ratio to have a constant expectation as well and conjecture its actual value from computer experiments. © 2015 Elsevier B.V.

Hypergraph Tur\'an Problems in ℓ2\ell_2-Norm

There are various different notions measuring extremality of hypergraphs. In this survey we compare the recently introduced notion of the codegree squared extremal function with the Tur\'an function, the minimum codegree threshold and the uniform Tur\'an density. The codegree squared sum $\textrm{co}_2(G)$ of a $3$-uniform hypergraph $G$ is defined to be the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $x,y$. In other words, this is the square of the $\ell_2$-norm of the codegree vector. We are interested in how large $\textrm{co}_2(G)$ can be if we require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This maximum value of $\textrm{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$ is called the codegree squared extremal function, which we denote by $\textrm{exco}_2(n,H)$. We systemically study the extremal codegree squared sum of various $3$-uniform hypergraphs using various proof techniques. Some of our proofs rely on the flag algebra method while others use more classical tools such as the stability method. In particular, we (asymptotically) determine the codegree squared extremal numbers of matchings, stars, paths, cycles, and $F_5$, the $5$-vertex hypergraph with edge set $\{123,124,345\}$. Additionally, our paper has a survey format, as we state several conjectures and give an overview of Tur\'an densities, minimum codegree thresholds and codegree squared extremal numbers of popular hypergraphs. We intend to update the arXiv version of this paper regularly.Comment: Invited survey for BCC 2022, comments are welcom

The Scarf complex and betti numbers of powers of extremal ideals

This paper is concerned with finding bounds on betti numbers and describing combinatorially and topologically (minimal) free resolutions of powers of ideals generated by a fixed number $q$ of square-free monomials. Among such ideals, we focus on a specific ideal $\mathcal{E}_q$, which we call {\it extremal}, and which has the property that for each $r\ge 1$ the betti numbers of ${\mathcal{E}_q}^r$ are an upper bound for the betti numbers of $I^r$ for any ideal $I$ generated by $q$ square-free monomials (in any number of variables). We study the Scarf complex of the ideals ${\mathcal{E}_q}^r$ and use this simplicial complex to extract information on minimal free resolutions. In particular, we show that ${\mathcal{E}_q}^r$ has a minimal free resolution supported on its Scarf complex when $q\leq 4$ or when $r\leq 2$, and we describe explicitly this complex. For any $q$ and $r$, we also show that $\beta_1({\mathcal{E}_q}^r)$ is the smallest possible, or in other words equal to the number of edges in the Scarf complex. These results lead to effective bounds on the betti numbers of $I^r$, with $I$ as above. For example, we obtain that pd$(I^r)\leq 5$ for all ideals $I$ generated by $4$ square-free monomials and any $r\geq 1$

Extremal Betti Numbers and Applications to Monomial Ideals

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs of classical criteria due to Hochster, Reisner, and Stanley.Comment: Minor revision. 15 pages, Plain TeX with epsf.tex, 8 PostScript figures, PostScript file available also at http://www.math.columbia.edu/~psorin/eprints/monbetti.p

From Monomials to Words to graphs

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal generated by \sigma(I) in the free monoid is finitely generated. Whether there exists an ordering such that is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.Comment: 27 pages, 2 postscript figures, uses gastex.st