Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.09830

A Counterexample to Nivat's Conjecture for a Non-Convex Window of Full Affine Span

Abhishek Khetan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality $8$ with full affine span, together with an $F$-tiling $T$, such that the orbit closure of $T$ in $\{0,1\}^{\mathbb{Z}^2}$ does not contain a $1$-periodic $F$-tiling. Since every $F$-tiling is a low-complexity configuration with respect to the window $\bar F := \{-a : a \in F\}$, this supplies a "non-degenerate" counterexample, in a strong sense, to Nivat's conjecture for non-convex windows. This answers, in the negative, a question of Kari and Moutot (2023) whether every such counterexample must be degenerate, in the sense that the probing window is contained in a coset of a proper finite-index sublattice. We complement this with a positive result: for every exact cluster $F$ of full affine span whose cardinality is the square of a prime, every $F$-tiling has a $1$-periodic $F$-tiling in its orbit closure. Together with Szegedy's theorem that every tiling by a cluster of prime cardinality is $1$-periodic, this shows that no cluster of fewer than $8$ cells can exhibit the phenomenon, with the possible exception of cardinality $6$, which we leave open.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Abhishek Khetan. 2026-07-18. A Counterexample to Nivat's Conjecture for a Non-Convex Window of Full Affine Span. https://arxiv.org/abs/2607.09830

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗
arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

math.CO↗