[Paper Review] The linear Tur\'an number of the k-fan

Zoltán Füredi, András Gyárfás|arXiv (Cornell University)|Oct 9, 2017

graph theory and CDMA systems6 references3 citations

TL;DR

This paper establishes the exact linear Turán number for the $k$-fan hypergraph, proving that the maximum number of edges in a $k$-uniform linear hypergraph on $n$ vertices without a $k$-fan subhypergraph is $n^2/k^2$, achieved if and only if $n \equiv 0 \pmod{k}$ and the hypergraph is a transversal design with $k$ groups. It further characterizes extremal configurations for $k=3$ and extends results to forbidden configurations like $P$ and $C_{14}$, showing that the linear Turán number remains $n^2/9$ for certain $n$ when these are excluded.

ABSTRACT

A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\\cal{F}}$ of hypergraphs, the linear Tur\\'an number ${\ m ex}_{\ m lin}(n,{\\cal{F}})$ is the maximum number of edges in a $k$-uniform linear hypergraph $\\mathcal H$ on $n$ vertices that does not contain any member of ${\\cal{F}}$ as a subhypergraph. For $k\\ge 2$ the $k$-fan $F^k$ is the $k$-uniform linear hypergraph having $k$ edges $f_1,\\dots,f_k$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. We prove the following extension of Mantel's theorem $${\ m ex}_{\ m lin}(n,F^k)\\le {n^2 / k^2}.$$ Moreover, $|{\\mathcal H}|=n^2/k^2$ holds if and only if $n\\equiv 0\\pmod k$ and $\\mathcal H$ is a transversal design on $n$ points with $k$ groups. We also study ${\ m ex}_{\ m lin}(n,{\\cal{F}})$ where $\\cal{F}$ is any subset of the three linear triple systems with four triples on at most seven points.

Motivation & Objective

• To determine the exact linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, F^k)$ for the $k$-fan hypergraph, a $k$-uniform linear hypergraph with $k$ edges sharing a common vertex and one additional edge intersecting all in distinct vertices.
• To characterize the extremal hypergraphs achieving this bound, showing they are precisely transversal designs with $k$ equal-sized groups when $n \equiv 0 \pmod{k}$.
• To extend the analysis to $k=3$, determining $\mathrm{ex}_{\mathrm{lin}}(n, F^3)$ for $n \equiv 2 \pmod{3}$ and identifying all extremal triple systems.
• To investigate the linear Turán number when multiple configurations—$F^3$, $P$, and $C_{14}$—are forbidden, showing that the extremal number remains $n^2/9$ for infinitely many $n$.

Proposed method

• Prove an upper bound of $n^2/k^2$ on $\mathrm{ex}_{\mathrm{lin}}(n, F^k)$ using extremal hypergraph theory and double counting arguments on edge intersections.
• Establish that equality holds if and only if the extremal hypergraph is a transversal design $T(n,k)$ with $k$ groups, using properties of linear hypergraphs and group partitioning.
• For $n = k(m+1) - 1$, prove $\mathrm{ex}_{\mathrm{lin}}(n, F^k) \leq m^2 + m$ using truncated transversal designs derived from $T(n+1,k)$.
• Use graph factorization techniques, particularly 1-factorizations of complete bipartite graphs $K_{2^m,2^m}$, to construct extremal triple systems avoiding $F^3$, $P$, and $C_{14}$.
• Apply the standard method of selecting a 3-partite subhypergraph containing at least $2/9$ of the edges to relate linear Turán numbers to classical Turán numbers.
• Leverage known results on Steiner triple systems and configurations to show that $\mathrm{ex}_{\mathrm{lin}}(n, \{F^3, P, C_{14}\}) = n^2/9$ for $n = 3 \cdot 2^m$.

Experimental results

Research questions

• RQ1 What is the exact value of the linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, F^k)$ for the $k$-fan hypergraph?
• RQ2 For which $n$ and $k$ is the extremal hypergraph achieving $\mathrm{ex}_{\mathrm{lin}}(n, F^k)$ a transversal design with $k$ groups?
• RQ3 Can the extremal hypergraphs for $\mathrm{ex}_{\mathrm{lin}}(n, F^3)$ be completely characterized when $n \equiv 2 \pmod{3}$?
• RQ4 What is the linear Turán number when multiple small configurations—$F^3$, $P$, and $C_{14}$—are forbidden simultaneously?
• RQ5 How do the linear Turán numbers for forbidden configurations relate to the classical Turán number for $F(7,4)$?

Key findings

• The linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, F^k)$ is bounded above by $n^2/k^2$ for all $k \geq 2$ and all $n$, with equality if and only if $n \equiv 0 \pmod{k}$ and the extremal hypergraph is a transversal design with $k$ groups.
• For $n = k(m+1) - 1$, the linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, F^k)$ is at most $m^2 + m$, with extremal hypergraphs being truncated transversal designs from $T(n+1,k)$.
• When $n = 3m + 2$, the extremal number $\mathrm{ex}_{\mathrm{lin}}(n, F^3)$ is exactly $m^2 + m$, and the only extremal triple systems are those derived from truncated transversal designs or specific graph extensions.
• For $n = 6m + 3$, the linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, \{F^3, P\})$ equals $n^2/9$, showing that forbidding $P$ does not reduce the extremal number beyond the $F^3$-free bound.
• For $n = 3 \cdot 2^m$, the linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, \{F^3, C_{14}\})$ is exactly $n^2/9$, achieved by a transversal design constructed from a 1-factorization of $K_{2^m,2^m}$.
• The linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, \{F^3, P, C_{14}\})$ is asymptotically equivalent to $n^2/9$, and this value is also the maximum for $\mathrm{ex}_{\mathrm{lin}}(n, \{P, C_{14}\})$ up to a constant factor, linking it to the classical Turán number for $F(7,4)$.