Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.32461

Optimally pseudorandom $K_4$-free graphs

Jie Han·Ferdinand Ihringer·Hendrik Van Maldeghem
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that optimally pseudorandom $K_4$-free graphs of order $n$ and degree $d = Θ(n^{4/5})$ exist by constructing a graph in the split Cayley hexagon, matching the known upper bound. This resolves the first open case for $K_k$-free graphs after $k=3$ for which Alon gave a tight construction in 1994. This has a variety of implications for $K_4$-free pseudorandom graphs. Furthermore, it implies an explicit lower bound on the Ramsey number of $r(4, t) \geq t^{1.\overline{6} - o(1)}$, improving the previous record by Kostochka, Pudlák, and Rödl of $r(4, t) \geq t^{1.6-o(1)}$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jie Han, Ferdinand Ihringer, Hendrik Van Maldeghem. 2026-09-26. Optimally pseudorandom $K_4$-free graphs. https://arxiv.org/abs/2609.32461

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

KEEP EXPLORING

Related papers

arXiv2026

The Rainbow Saturation Number of Cycles

An edge-colored graph $(G,\mathcal C)$ is $F$-rainbow saturated if it contains no rainbow copy of $F$, but adding any nonedge in any prescribed color creates one. The rainbow saturation number $\operatorname{rsat}(n,F)$ is the minimum number of edges in such a graph of order $n$. In this paper we study $\operatorname{rsat}(n,C_r)$. We determine the value for $C_4$ exactly, and establish bounds for every fixed $C_r$ with $r\geq5$.

math.CO↗
arXiv2026

Moreira's Theorem for Image Partition Regular Matrices

The famous Hindman conjecture says that for any finite coloring of natural numbers, there exists a monochromatic copy of the form $\{x,x+y,xy\}.$ In a celebrated article, Moreira gave a partial answer to this conjecture by showing that every finite coloring of the natural numbers contains a monochromatic configuration of the form $\{x, x+y, xy\}$. In this article we prove matrix versions (both finite and infinite) of Moreira's theorem. A matrix $A$ is said to be an image partition regular matrix if for any finite coloring of naturals, there exists a monochromatic image of $A,$ i.e. there exists a vector $\vec X$ such that all the entries of $A\vec X$ are monochromatic. From a recent paper of Bowen, one can derive the finite matrix version of the Moreira theorem: if $A$ and $B$ are two finite image partition regular matrices of the same order, then under any finite coloring of $\mathbb{N}$, there exist vectors $\vec{X}$ and $\vec{Y}$ such that all entries in the union of $A\vec{X}, A\vec{X} + B\vec{Y}, A\vec{X} \cdot B\vec{Y}$ are monochromatic, where $A\vec{X} \cdot B\vec{Y}$ denote the vector each of its entries are pointwise multiplication of the coordinates of $A\vec{X} \text{ and } B\vec{Y}$. In this article, we give a short combinatorial proof of this result, and then we extend it to infinite image partition regular matrices.

math.CO↗
arXiv2026

Explicit enumeration and large-valence asymptotics of even-valent maps

Let $\mathscr{N}_g(2ν,j)$ denote the number of connected labelled $2ν$-valent maps of genus $g$ with $j$ vertices. Explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ have been available only in the planar and toroidal cases. Using a structural formula of Ercolani et al (2023), we translate the problem of determining an explicit bivariate formula for $\mathscr{N}_g(2ν,j)$, $g \geq 2$, to finding finitely many counts with a fixed number of vertices. For $g=2,3$ and $4$ we determine these counts using the associated orthogonal polynomials, yielding explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ in these genera. Furthermore, the same method applies for every $g\ge5$ at the cost of additional computation. From these formulae we obtain the leading-order asymptotics of $\mathscr{N}_g(2ν,j)$ as $ν\to\infty$ for $g=2,3,4$, and we conjecture the structure of these formulae in general genus. In addition, we establish an analogous reduction to finitely many counts, derive explicit formulae and large-valence asymptotics, and formulate corresponding conjectures for two-legged even-valent maps.

math.CO↗