On almost Gallai colourings in complete graphs

TL;DR

The paper studies almost $t$-Gallai colourings of complete graphs, where rainbow $t$-cliques are allowed only if edge-disjoint. For $t\ge 4$, it proves subquadratic growth $\tau_t(n)=o(n^2)$ with a matching $n^{2-o(1)}$ lower bound, using a random $t$-partition construction and the Clique Removal Lemma, alongside a Kovács–Nagy-type lower-bound construction. For $t=3$, it gives the first nontrivial narrowing of bounds, showing $(\tfrac12 - o(1))n\log n \le \tau_3(n) = O(n^{\sqrt{2}}\log n)$, via a two-pronged approach combining a careful 3-colouring analysis with hypercube-edge isoperimetric arguments and probabilistic methods. A lower bound for $t=3$ uses an explicit almost Gallai colouring built on a hypercube embedding, achieving about $n\log n$ rainbow triangles, while the upper bound hinges on balanced/unbalanced case analyses and a decoupling trick to control the distribution of rainbow cliques. The results yield anticoncentration bounds for clique counts in $G(n,p)$ and advance the understanding of how structural constraints on rainbow subgraphs constrain extremal counts in both deterministic and random settings.

Abstract

For $t \in \mathbb{N}$, we say that a colouring of $E(K_n)$ is $\textit{almost}$ $t$-$\textit{Gallai}$ if no two rainbow $t$-cliques share an edge. Motivated by a lemma of Berkowitz on bounding the modulus of the characteristic function of clique counts in random graphs, we study the maximum number $τ_t(n)$ of rainbow $t$-cliques in an almost $t$-Gallai colouring of $E(K_n)$. For every $t \ge 4$, we show that $n^{2-o(1)} \leq τ_t(n) = o(n^2)$. For $t=3$, surprisingly, the behaviour is substantially different. Our main result establishes that $$\left ( \frac{1}{2}-o(1) \right ) n\log n \le τ_3(n) = O\big (n^{\sqrt{2}}\log n \big ),$$ which gives the first non-trivial improvements over the simple lower and upper bounds. Our proof combines various applications of the probabilistic method and a generalisation of the edge-isoperimetric inequality for the hypercube.

Figures (3)

• Figure 1: The colouring for $m = 2$. Only the edges of rainbow triangles are shown.
• Figure 2: A nice quadruple of vertices.
• Figure 3: Partition of $V_2$ and $V_3$ into the red and blue neighbourhoods of $v$, and the red neighbourhood of $u$ inside $V_{2,B}$ and $V_{3,B}$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Remark
• Theorem 1.3
• Lemma 4.1
• proof
• Lemma 4.2
• Lemma 4.3
• proof : Proof of the upper bound in Theorem \ref{['th:rainbow_triangles']}
• Lemma 5.1