The authors study linear hypergraphs, which have edges that share at most one vertex, focusing on how many edges such hypergraphs can have without containing certain forbidden small subhypergraphs called hypertrees. They determine exact or close-to-exact maximum edge counts for various types of structured hypertrees, including stars, brooms, crowns, and paths, often linking these results to special combinatorial designs called Steiner systems. For example, they prove sharp bounds for linear stars and characterize when those bounds are tight, and they provide constructions that come close to or match these bounds for other hypertrees. They also identify errors in previous work for 4-uniform linear paths and offer corrected proofs and characterizations. Overall, their results extend previous knowledge from 3-uniform to higher uniformities in linear hypergraphs.
Authors
Rajat Adak, Pragya Verma
Abstract
A hypergraph is \emph{linear} if every pair of vertices is contained in at most one hyperedge. For a family $\mathcal{F}$ of $r$-uniform hypergraphs, the linear Tur'an number $ex^{\mathrm{lin}}_r(n,\mathcal{F})$ is the maximum number of hyperedges in an $n$-vertex $\mathcal{F}$-free linear $r$-uniform hypergraph. Extending the work of Gy'arf'as, Ruszink'o, and S'ark"ozy on $3$-uniform linear hypertrees, we study linear Tur'an numbers for higher uniformity. We determine the linear Tur'an number of the $r$-uniform linear star $S_k^r$, proving [ex^{\mathrm{lin}}_r(n,S_k^r)\le \frac{n(k-1)}{r},] with equality exactly for $(k-1)$-regular linear $r$-uniform hypergraphs, whenever they exist. We also construct dense $T_k^r$-free hypergraphs showing that, under suitable divisibility and design-existence assumptions, [ex^{\mathrm{lin}}_r(n,T_k^r)\ge \frac{n(k-1)}{r}] for every linear $r$-uniform hypertree $T_k^r$ with $k$ hyperedges. We then study all linear hypertrees with four hyperedges. For the broom $B_4^r$, we prove [ex^{\mathrm{lin}}_r(n,B_4^r)\le \frac{(r+1)n}{r},] and characterize the extremal hypergraphs as disjoint unions of Steiner systems $S(2,r,r^2)$, whenever such systems exist. For the crown $E_4^r$, we establish [ex^{\mathrm{lin}}_r(n,E_4^r)\le \frac{(2r-1)n}{r},] together with a lower-bound construction leaving only a constant-factor gap. For the linear path $P_4^r$, we construct $P_4^r$-free hypergraphs with $(r+1)n/r$ hyperedges and conjecture this is the optimal general bound. We verify the conjecture for connected hypergraphs under suitable degree conditions. Finally, for $r=4$, we identify counterexamples to a key structural claim in a previous proof of Zhang and Wang, give a new proof that [ex^{\mathrm{lin}}_4(n,P_4^4)\le \frac{5n}{4},] and show that equality holds precisely for disjoint unions of Steiner systems $S(2,4,16)$.