# Singular Ramsey and Tur\'an numbers

## TL;DR

This paper introduces and studies the concepts of singular Ramsey and Turán numbers, providing methods to estimate them and establishing tight bounds and exact results for these new extremal graph parameters.

## Contribution

It initiates the study of singular Ramsey and Turán numbers, developing estimation methods and deriving tight bounds and exact values.

## Key findings

- Developed methods to estimate Rs(F) and Ts(n,F)
- Established tight asymptotic bounds for the numbers
- Provided exact results for specific cases

## Abstract

We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $v\in V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $m\geq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Tur\'an number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs$(F)$ and Ts$(n,F)$, present tight asymptotic bounds and exact results.