[Paper Review] Generalizations of the Ruzsa-Szemer\'edi and rainbow Tur\'an problems for cliques

W. T. Gowers, Barnabás Janzer|arXiv (Cornell University)|Mar 5, 2020

Limits and Structures in Graph Theory12 references4 citations

TL;DR

This paper generalizes the Ruzsa–Szemerédi and rainbow Turán problems to cliques, proving that for any r < s, there exist graphs on n vertices with n^{r−o(1)} copies of K_s such that each K_r is contained in at most one K_s. It also establishes that ex(n, K_r, rainbow-K_r) = n^{r−1−o(1)} for r ≥ 4, resolving a question on the order of magnitude of the generalized rainbow Turán number for cliques.

ABSTRACT

Considering a natural generalization of the Ruzsa-Szemer\'edi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $n$ vertices containing $n^{r}e^{-O(\sqrt{\log{n}})}=n^{r-o(1)}$ copies of $K_s$ such that any $K_r$ is contained in at most one $K_s$. We also give bounds for the generalized rainbow Tur\'an problem $\operatorname{ex}(n, H,$rainbow-$F)$ when $F$ is complete. In particular, we answer a question of Gerbner, M\'esz\'aros, Methuku and Palmer, showing that there are properly edge-coloured graphs on $n$ vertices with $n^{r-1-o(1)}$ copies of $K_r$ such that no $K_r$ is rainbow.

Motivation & Objective

• To generalize the Ruzsa–Szemerédi problem to higher cliques, determining the maximum number of K_s copies in a graph where each K_r is in at most one K_s.
• To resolve the order of magnitude of the generalized rainbow Turán number ex(n, K_r, rainbow-K_r) for r ≥ 4.
• To develop geometric and product-based constructions that achieve near-optimal bounds in the generalized clique setting.
• To answer a question posed by Gerbner, Mészáros, Methuku, and Palmer on the asymptotic growth of ex(n, K_r, rainbow-K_r).

Proposed method

• Constructs a geometric graph using points in R^d with specific vector assignments to ensure controlled clique containment.
• Uses a 6-partite construction with vertex permutations and edge colouring automorphisms to generate multiple copies of K_6 with no rainbow K_4.
• Applies a product construction across multiple copies of the base graph to amplify the number of K_6 copies while preserving the rainbow-free property.
• Employs a coloring scheme on K_6 with 5 colors and symmetry properties (automorphism group acting transitively) to maintain structural consistency across copies.
• Leverages the concept of c-spanning sets and subspace dimension to bound the number of K_4-free configurations in king’s graphs H_{k,l}.
• Uses vector representations of vertices to model edge colorings and apply linear algebraic conditions ensuring no rainbow K_4 appears.

Experimental results

Research questions

• RQ1 What is the maximum number of K_s copies in a graph on n vertices such that each K_r is contained in at most one K_s, for fixed r < s?
• RQ2 Can the lower bound construction for the Ruzsa–Szemerédi problem be generalized to higher cliques r ≥ 3?
• RQ3 What is the order of magnitude of ex(n, K_r, rainbow-K_r) for r ≥ 4, and is the upper bound of O(n^{r−1}) sharp?
• RQ4 Can geometric or product-based constructions achieve near-optimal exponents in generalized Turán problems for cliques?
• RQ5 What is the asymptotic behavior of ex(n, H_{k,l}, rainbow-K_4) for king’s graphs H_{k,l}?

Key findings

• For any fixed r < s, there exists a graph on n vertices with n^{r−o(1)} copies of K_s such that every K_r is contained in at most one K_s, establishing a nearly optimal lower bound.
• The construction achieves n^{r−o(1)} copies of K_s, matching the exponent in the Ruzsa–Szemerédi case (r=2, s=3) but generalized to higher cliques.
• It is shown that ex(n, K_r, rainbow-K_r) = n^{r−1−o(1)} for all r ≥ 4, resolving the open question on the order of magnitude of this generalized rainbow Turán number.
• A product construction using 6-partite graphs and color-preserving permutations yields at least n^{12−o(1)} copies of K_6 with no rainbow K_4, improving the base construction.
• For king’s graphs H_{k,l}, the bound ex(n, H_{k,l}, rainbow-K_4) = n^{k+l−1−o(1)} is established, showing the exponent matches the dimension of the underlying vector space.
• The paper confirms that the upper bound of O(n^{r−1}) for ex(n, K_r, rainbow-K_r) is sharp, as the lower bound matches n^{r−1−o(1)}.