Ramsey-Tur\'an numbers for intersecting odd cliques

TL;DR

This paper determines the asymptotic maximum number of edges in large graphs avoiding certain intersecting odd cliques and independent sets, extending classical Ramsey-Turán results to more complex graph structures.

Contribution

It extends the classical Ramsey-Turán theorem to graphs containing multiple intersecting odd cliques sharing a vertex, showing they have similar edge bounds as single odd cliques.

Findings

The maximum edges in graphs avoiding intersecting odd cliques match classical bounds.

The results generalize Erdős and Sós's 1969 theorem.

Asymptotic edge bounds are established for complex clique structures.

Abstract

Given a graph $H$ and a function $f:\mathbb{Z}^+ \longrightarrow \mathbb{Z}^+ $, the Ramsey-Tur\'an number of $H$ and $f$, denoted by $RT(n, H, f(n))$, is the maximum number of edges a graph $G$ on $n$ vertices can have, which does not contain $H$ as a subgraph and also does not contain a set of $f(n)$ independent vertices. Let $r$ be a positive integer. In 1969, Erd\H{o}s and S\'os proved that $RT(n,K_{2r+1},o(n))=\frac{n^2}{2}(1-\frac{1}{r})+o(n^2)$. Let $F_k(2r+1)$ denote the graph consisting of $k$ copies of complete graphs $K_{2r+1}$ sharing exactly one vertex. In this paper, we show that $RT(n,F_k(2r+1),o(n))=\frac{n^2}{2}(1-\frac{1}{r})+o(n^2)$, which is of the same magnitude with $RT(n, K_{2r+1}, o(n))$.