arXiv subjects

Pedram Noohi

Publications and source records attributed to Pedram Noohi.

2 recordsLinked to original sources

arXiv2026

Hypergraphs with the smallest Hall ratio are still hard to colour

For $H$ a hypergraph, the Hall ratio $ρ(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{α(H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k = 2$), having minimum Hall ratio $2$ forces chromatic number $2$. Nešetřil, Rödl, and Sales asked whether a similar phenomenon holds at higher uniformities. We answer this question negatively in a strong form, showing that for every $k \geqslant 3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.

math.CO↗

arXiv2026

Decay-Ramsey Sequences

Given a sequence $\mathbf{a}=(a_i)_{i\geq 1}$ of positive real numbers, we say that a colouring $c:E(K_n)\to \mathbb{N}$ is $\mathbf{a}$-bounded if the proportion of edges coloured $i$ is at most $a_i$, for all $i$. We write \[ \mathbf{a}\, \to\, K_k \] if, for all sufficiently large $n$, all $\mathbf{a}$-bounded colourings $c:E(K_n)\to \mathbb{N}$ contain a monochromatic $K_k$. We say that the sequence $\mathbf{a}$ is \emph{Decay-Ramsey} if $\mathbf{a}\to K_k$ for all $k\ge 2$. In this short paper we introduce the question of determining which sequences are Decay-Ramsey and present some initial results.

math.CO↗