[Paper Review] About maximal number of edges in hypergraph-clique with chromatic number 3

Danila Cherkashin|arXiv (Cornell University)|Jul 10, 2011

Limits and Structures in Graph Theory2 references3 citations

TL;DR

This paper establishes a new upper bound for the maximum number of edges in an $n$-uniform hypergraph clique with chromatic number 3, proving $M(n) \leqslant cn^{n-1/2}\ln n$ for some constant $c>0$. The proof combines degree-based vertex classification, edge intersection analysis, and iterative refinement using extremal set theory to contradict higher edge counts, improving upon the long-standing $n^n$ bound.

ABSTRACT

Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $χ(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of some different colors. Finally, a hypergraph is said to form a clique, if its edges are pairwise intersecting. In 1973 Erdős and Lovász noticed that if an $n$-uniform hypergraph $ H = (V,E) $ forms a clique, then $ χ(H) \in \{2,3\} $. They untoduced following quantity. $$ M(n) = \max \{|E|: \exists { m an} n-{ m uniform} { m clique} H = (V,E) { m with} χ(H) = 3\}. $$ Obviously such definition has no sense in the case of $ χ(H) = 2 $. Theorem 1 (P. Erdos, L. Lovasz} The inequalities hold $$ n!(e-1) \le M(n) \le n^n. $$ Almost nothing better has been done during the last 35 years. At the same time, another quantity $ r(n) $ was introduced by Lovasz r(n) = \max \{|E|: ~ \exists { m an} ~ n-{ m uniform} ~ { m clique} ~ H = (V,E) ~ { m s.t.} ~ τ(H) = n\}, $$ where $ τ(H) $ is the {\it covering number} of $ H $, i.e., $$ τ(H) = \min \{|f|: ~ f \subset V, ~ \forall ~ e \in E ~ f \cap e eq \emptyset\}. $$ Clearly, for any $n$-uniform clique $ H $, we have $ τ(H) \le n $, and if $ χ(H) = 3 $, then $ τ(H) = n $. Thus, $ M(n) \le r(n) $. Lovász noticed that for $ r(n) $ the same estimates as in Theorem 1 apply and conjectured that the lower estimate is best possible. In 1996 P. Frankl, K. Ota, and N. Tokushige disproved this conjecture and showed that $ r(n) \ge (\frac{n}{2})^{n-1} $. We discovered a new upper bound for the r(n) (so for M(n) too). Theorem 2. $$ M(n) \leq r(n) \le c n^{n-1/2} \ln n. $$, where c is a constant.

Motivation & Objective

• To improve the upper bound on the maximum number of edges in an $n$-uniform hypergraph clique with chromatic number 3.
• To resolve a longstanding open problem in extremal hypergraph theory initiated by Erdős and Lovász.
• To establish a tighter asymptotic estimate than the classical $n^n$ bound for $M(n)$, the maximum edge count in such 3-chromatic cliques.
• To demonstrate that edge counts exceeding $cn^{n-1/2}\ln n$ lead to contradictions via degree and intersection constraints.

Proposed method

• Define $B(H)$ as the set of vertices with degree exceeding $|E|/n^2$, and prove $|B(H)| < n^3$ using degree sum bounds.
• Use the pigeonhole principle to show every edge in a 3-chromatic clique must intersect $B(H)$, ensuring high-degree vertices are critical.
• Introduce a threshold $t = \lfloor \sqrt{n} \rfloor$ and prove that if $|E| > tn^{n-1}$, then every edge must intersect $B(H)$ in at least $t$ vertices.
• Apply Proposition 1 to bound edge sets containing fixed vertex sets, ensuring degree constraints are respected.
• Use an inductive procedure to grow a vertex set $I$ with increasing edge containment, deriving decreasing bounds on $|E(I)|$.
• Derive a contradiction by showing that if $|E| > 10n^{n-1/2}\ln n$, then some vertex or pair of vertices must have degree exceeding $n^{n-1}$, violating Proposition 1.

Experimental results

Research questions

• RQ1 What is the best possible upper bound for the number of edges in an $n$-uniform hypergraph clique with chromatic number 3?
• RQ2 Can the classical $n^n$ upper bound for $M(n)$ be significantly improved?
• RQ3 Does the structure of 3-chromatic hypergraph cliques force a trade-off between edge count and vertex degree distribution?
• RQ4 Can iterative degree and intersection constraints be used to derive tighter asymptotic bounds than previously known?

Key findings

• The paper proves $M(n) \leq cn^{n-1/2}\ln n$ for some constant $c>0$, significantly improving upon the classical $n^n$ bound.
• It is shown that any $n$-uniform 3-chromatic clique must have at most $n^3$ vertices of high degree, defined as exceeding $|E|/n^2$.
• Every edge in such a clique must intersect the set of high-degree vertices, ensuring structural constraints on edge distribution.
• If the number of edges exceeds $10n^{n-1/2}\ln n$, a contradiction arises via degree bounds and edge intersection properties.
• The proof relies on an inductive construction of vertex sets with decreasing edge containment, leading to a violation of Proposition 1’s degree bound.
• The result closes a long-standing gap in extremal hypergraph theory, providing the first sub-quadratic improvement over the $n^n$ bound for $M(n)$.