# On the Profile of Multiplicities of Complete Subgraphs

## TL;DR

This paper establishes new lower bounds on the number of monochromatic complete subgraphs of various sizes in large 2-colored complete graphs, improving previous bounds and matching known upper bounds under certain assumptions.

## Contribution

It provides improved lower bounds for the count of monochromatic complete subgraphs of specific sizes in large 2-colored complete graphs, and characterizes their asymptotic behavior.

## Key findings

- Lower bound of $n^{(rac{1}{4} - o(1)) ext{log} n}$ for subgraphs of size between $0.3 ext{log} n$ and $0.7 ext{log} n$.
- At least $n^{rac{1}{7} ext{log} n}$ monochromatic subgraphs of size $rac{1}{2} ext{log} n$.
- Matching bounds for the number of subgraphs of size $c ext{log} n$ assuming the largest monochromatic subgraph size is about $(rac{1}{2}+o(1)) ext{log} n$.

## Abstract

Let $G$ be a $2$-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} - o(1))\log n}$ monochromatic complete subgraphs of size $r$, where \[ 0.3\log n < r < 0.7\log n. \] The previously known lower bound on the total number of monochromatic complete subgraphs, due to Sz\'{e}kely was $n^{0.1576\log n}$. We also prove that $G$ contains at least $n^{\frac{1}{7} \log n} $ monochromatic complete subgraphs of size $\frac{1}{2}\log n$. If furthermore one assumes that the largest monochromatic complete subgraph in $G$ is of size $(\frac{1}{2} + o(1))\log n$ (it is a well known open question whether such graphs exist), then for every constant $0 \le c \le \frac{1}{2}$ we determine (up to low order terms) the number of monochromatic complete subgraphs of size $c \log n$. We do so by proving a lower bound that matches (up to low order terms) a previous upper bound of Sz\'{e}kely. For example, the number of monochromatic complete subgraphs of size $\frac{1}{2} \log n$ is $n^{\frac{1}{8}(4 - \log e \pm o(1))\log n} \simeq n^{0.32 \log n}$.