Counting cliques with prescribed intersection sizes
Yuhao Zhao, Xiande Zhang
TL;DR
This work resolves the asymptotic behavior of the generalized Turán problem for cliques with restricted intersections. It establishes a sharp dichotomy: $Ψ_r(n,L)$ grows polynomially of degree $|L|$ if and only if the endpoints $\ell_1,\dots,\ell_s,r$ form an arithmetic progression; in that case, the paper derives exact asymptotics $Ψ_r(n,L) = (1+o(1))\left(\frac{n-\ell_1}{r-\ell_1}\right)^{|L|}$ and proves the extremal graph is unique up to isomorphism, realized by a blow-up of a Turán graph with a fixed clique. In the non-AP case, $Ψ_r(n,L)=o(n^{|L|})$, and the AP case leads to a constructive extremal model $G_{n,r,L}$. The authors also prove a Hilton–Milner-type stability for the $(K_r,t)$-intersecting regime and discuss further avenues, including cover-free and Erdős–matching-type generalizations, highlighting rich connections between extremal graph theory and extremal set theory.
Abstract
We study the generalized Turán problem regarding cliques with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subset [0,r-1]$ be a fixed integer set with $|L|\notin \{1,r\}$ and $\ell_1<\dots<\ell_s$, and let $Ψ_r(n,L)$ denote the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques are $L$-intersecting as a family of $r$-subsets. Helliar and Liu recently initiated the systematic study of the function $Ψ_r(n,L)$ and showed that $Ψ_r(n,L)\le \left(1-\frac{1}{3r}\right) \prod_{\ell\in L}\frac{n-\ell}{r-\ell}$ for large $n$, improving the trivial bound from the Deza--Erdős--Frankl theorem by a factor of $1-\frac{1}{3r}$. In this article, we improve their result by showing that as $n$ goes to infinity $Ψ_r(n,L)=Θ_{r,L}(n^{|L|})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and fully determining the corresponding exact values of $Ψ_r(n,L)$ for sufficiently large $n$ in this case. Moreover, when $L=[t,r-1]$, for the generalized Turán extension of the Erdős--Ko--Rado theorem given by Helliar and Liu, we show a Hilton--Milner-type stability result.