A note on the $t$-partite link problem of F\"uredi

TL;DR

This paper establishes an upper bound on the maximum edge density of 3-graphs with t-partite link graphs, matching known lower bounds up to a constant factor, thus nearly determining the asymptotic behavior.

Contribution

It proves a new upper bound on the edge density of 3-graphs with t-partite link graphs, extending previous constructions and bounds to a more general setting.

Findings

Upper bound $oxed{ ext{pi}_{ ext{link}}(t) extless= 1 - t^{-1} - t^{-2}/12}$ for all $t extgreater= 2$.

Matching the upper bound with Goldwasser's lower bound up to a constant factor for prime-power $t-1$.

Extension of the bound to 3-graphs with no generalized daisies and $K_{t+1}$-free link graphs.

Abstract

Motivated by the Erd\H{o}s--S\'{o}s bipartite link conjecture, F\"{u}redi (Oberwolfach, 2004) asked for the asymptotic maximum edge density $\pi_{\mathrm{link}}(t)$ of $3$-graphs in which the link graph of every vertex is $t$-partite. Goldwasser's recursive blow-up construction based on projective planes gives the lower bound $\pi_{\mathrm{link}}(t)\ge 1-t^{-1}-(2+o_t(1))t^{-2}$ whenever $t-1$ is a prime power. In this note, we prove the upper bound $\pi_{\mathrm{link}}(t)\le 1-t^{-1}-t^{-2}/12$ for every $t \ge 2$. Together with Goldwasser's construction, this determines, up to a constant factor, the correct order of the gap between $\pi_{\mathrm{link}}(t)$ and the trivial averaging upper bound $1-t^{-1}$ for all prime-power values of $t-1$. In fact, our argument applies in the more general setting of $3$-graphs with no generalized daisies, equivalently, $3$-graphs in which the link graph of every vertex is $K_{t+1}$-free. We also establish an analogous upper bound for the positive $(r-1)$-codegree Tur\'an density of generalized daisies.