Blocking Amalgamations, Maximal Arcs, and Generalized Crowns

Abstract: Let $C<sup>r_{1,k}$ be the $r$-uniform $k$-crown and put $h=r-k+2$. For a finite linear intersecting $r$-uniform hypergraph $G$, let $τ<em>h(G)$ be the minimum size of a set meeting every edge of $G$ in at least $h$ vertices, and define [ ρ{r,k}=\sup_G\frac{|E(G)|}{τh(G)}. ] We prove that every fixed pair $(G,B)$, with $B$ an $h$-fold transversal, yields [ \operatorname{ex}^{\mathrm{lin}}_r(n,C^r{1,k}) \ge \frac{|E(G)|}{|B|}n-O_{G,B}(\sqrt n) ] for all sufficiently large $n$. Incidence counting gives $ρ<em>{r,k}\le r/h$, and equality is characterized after dualization by a pairwise balanced design with a distinguished regular subfamily. For $r=q+1$, where $q$ is a prime power, truncated projective planes give [ \frac qh\le ρ{q+1,k}\le\frac{q+1}{h}. ] The upper endpoint is attained whenever a maximal $h$-arc exists; in particular, if $q$ is even and $h\mid q$, then $ρ<em>{q+1,k}=(q+1)/h$. Padding the truncated-plane construction gives [ ρ{r,r}=(1-o(1))\frac r2 ] and, uniformly for each fixed $\varepsilon&gt;0$ and $\varepsilon r\le k\le r$, [ ρ_{r,k}=(1+o(1))\frac{r}{r-k+2}. ] For nonintersecting templates, the corresponding transfer is governed by a local safe-block condition that replaces the $h$-fold transversal requirement.

• The paper develops constrained-port amalgamations that produce linear, generalized-crown-free hypergraphs with asymptotic edge density at least the efficiency parameter \(\rho_{r,k}\), including an explicit \(O(\sqrt n)\) error term.
• It proves the universal bound \(\rho_{r,k}\le r/(r-k+2)\) and shows equality is rigid, corresponding to pairwise balanced designs and maximal arcs in projective planes.
• Projective-plane truncation, maximal-arc constructions, and padding nearly attain the bound for arbitrary uniformities and crowns of linear size, while safe blockers identify open routes beyond intersecting templates.

This paper develops a systematic theory of block-amalgamation constructions for linear Turán problems concerning generalized crowns $C^r_{1,k}$, the $r$-uniform hypergraphs consisting of a base edge with $k$ pairwise disjoint petals attached at distinct base vertices. The central object is the efficiency parameter

$$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$

where the supremum ranges over finite linear intersecting $r$-uniform hypergraphs and $\tau_h(G)$ is the minimum size of an $h$-fold transversal (a set meeting every edge in at least $h$ vertices). The paper proves that $\rho_{r,k}$ governs explicit lower bounds on $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$, establishes the universal ceiling $r$0, characterizes equality via maximal arcs in projective planes, and shows that the ceiling is asymptotically attained for every crown size linear in the uniformity.

The constrained-port amalgamation

The construction separates two roles within a template block $r$1. A distinguished set $r$2 of vertices remains private to each copy of $r$3, while the remaining vertices are shared across copies through port maps. The key combinatorial device is the port-shadow graph $r$4 on the shared vertices $r$5, where $r$6 is an edge precisely when some edge of $r$7 contains both $r$8 and $r$9. A family of port maps $k$0 is $k$1-constrained if for every shadow edge $k$2, the pair map $k$3 is injective; this generalizes mixed-alphabet packing arrays of strength two (2608.16035).

The main transfer theorem states that for any linear intersecting $k$4 with an $k$5-fold transversal $k$6 and any constrained port array, the amalgamation is linear and $k$7-free, with exactly $k$8 edges and at most $k$9 vertices. Crown-freeness follows from a simple counting argument: since $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$0 meets every edge in at least $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$1 vertices, at most $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$2 base vertices are shared, so any putative $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$3-crown would attach at least two petals at private vertices, forcing both petals into one copy of the intersecting block — where they cannot be disjoint. Instantiating the port array via parallel classes of an affine plane $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$4 with $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$5 yields the all-orders bound

$$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$6

using Bertrand's postulate to select a prime affine order near $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$7. Consequently $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$8, so the parameter genuinely controls achievable densities for all large $$\rho_{r,k}=\sup_G \frac{|E(G)|}{\tau_h(G)},\qquad h=r-k+2,$$9, not merely along subsequences.

The efficiency ceiling and its rigidity

Incidence counting gives the universal upper bound $r$0. The proof first establishes that a non-star intersecting linear $r$1-uniform hypergraph has maximum degree at most $r$2: a vertex of degree exceeding $r$3 would force every edge through it, making the hypergraph a star. Stars are then handled separately, with ratio strictly below $r$4. In the non-star case the paper derives an exact defect identity,

$r$5

showing that equality holds if and only if every blocker vertex has degree $r$6 and every edge meets $r$7 in exactly $r$8 vertices. This identity records every source of inefficiency explicitly, which is stronger than a bare inequality.

Dualization reveals why finite geometry enters. For an extremal pair $r$9, the sets $\tau_h(G)$0 form a pairwise balanced design on the point set $\tau_h(G)$1 in which every design point has replication number $\tau_h(G)$2, every block has size at most $\tau_h(G)$3, and the blocks indexed by $\tau_h(G)$4 all have size $\tau_h(G)$5 with each design point lying in exactly $\tau_h(G)$6 of them. Moreover $\tau_h(G)$7, and if $\tau_h(G)$8 then necessarily $\tau_h(G)$9 and adjoining one edge to $h$0 produces a projective plane of order $h$1, with $h$2 the complement of the adjoined line. Equality configurations are thus rigidly geometric.

Maximal arcs attain the ceiling

Within a projective plane $h$3 of order $h$4, incidence counting shows that $h$5 for any line subfamily $h$6 and $h$7-fold transversal $h$8, with equality if and only if $h$9 is a maximal arc of degree $h$0 (a point set meeting every line in $h$1 or $h$2 points) and $h$3 is its secant set. Since secant-line counts satisfy $h$4 and the ceiling forces $h$5, maximal arcs yield exact optimizers:

• $h$6 for every prime power $h$7, using the complement of a line as a degree-$h$8 maximal arc;
• $h$9 whenever $\rho_{r,k}$0 is even and $\rho_{r,k}$1 divides $\rho_{r,k}$2, by Denniston's construction (2608.16035);
• $\rho_{r,k}$3 for every even prime power $\rho_{r,k}$4.

For odd $\rho_{r,k}$5, nontrivial maximal arcs do not exist in Desarguesian planes (2608.16035), so the exact value remains open there; the paper is careful to state this dependence on the arithmetic of $\rho_{r,k}$6 rather than claiming full generality.

Truncated planes and arbitrary uniformities

For parameters outside the divisibility range, deleting a point $\rho_{r,k}$7 of $\rho_{r,k}$8 together with all lines through it produces the truncated plane $\rho_{r,k}$9, which is dual to an affine plane and satisfies $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$0 for every $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$1. This gives the sandwich

$\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$2

with multiplicative gap $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$3 independent of $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$4 — equivalently, $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$5 uniformly over $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$6. The port-shadow structure here is transparent: choosing $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$7 partite classes as the blocker leaves a complete $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$8-partite shadow with chromatic number $\mathrm{ex}^{\mathrm{lin}}_r(n,C^r_{1,k})$9, far cheaper than assigning a direction to each of the $r$00 individual ports.

Padding lifts these results beyond uniformity $r$01: adjoining $r$02 private degree-one leaves to each edge preserves the ratio $r$03 whenever $r$04. Taking $r$05, the largest prime below $r$06, and invoking the prime number theorem yields the asymptotic optimization

$r$07

the latter uniformly over $r$08 for each fixed $r$09. These statements optimize the intersecting-block method asymptotically in every regime where the crown size is linear in the uniformity. Notably, the unrestricted Turán coefficient may be larger: Adak's upper bound gives $r$10 (Adak, 12 Apr 2026), leaving a factor-of-two-scale gap against the block-optimal value $r$11 at the full-crown endpoint.

Finite examples and a rank obstruction

Two concrete consequences illustrate the reach of the machinery. First, using three nonconcurrent lines of $r$12 as a double-transversal blocker and four ports colored by directions of $r$13, the paper constructs a connected linear $r$14-uniform full-crown-free hypergraph with $r$15 vertices and $r$16 edges, achieving density $r$17 — refuting the coefficient $r$18 in uniformity four even among connected examples.

Second, the amalgamation mechanism obstructs global incidence-rank inequalities. For every prime power $r$19, there exists a finite linear full-crown-free hypergraph $r$20 with $r$21, because crown-free densities approach at least $r$22 while rank is bounded by vertex count. The paper attributes this failure to global port sharing: projective geometry describes local equality configurations, but identifications among many copies raise the global density. Any endpoint argument based on incidence rank must therefore account for amalgamation.

Nonintersecting templates and safe blockers

The intersecting hypothesis enters the transfer theorem only once — forcing two privately attached petals into a common copy. For general linear templates, the correct local quantity is $r$23, the maximum number of pairwise disjoint edges meeting $r$24 at distinct points of $r$25. A set $r$26 is $r$27-safe if $r$28 for every edge. The transfer theorem extends verbatim under this condition, and conversely, in the canonical affine construction with $r$29, any unsafe blocker provably produces a crown via a greedy petal-selection argument. Hence safe-block templates yield lower bounds $r$30 on the liminf.

Crucially, within intersecting templates the relaxation does not help: the same ceiling $r$31 holds for all intersecting safe pairs, proved by a refined incidence count that charges degree-one vertices on nearly-covered edges. This identifies a concrete frontier — any construction beating $r$32 must use a genuinely nonintersecting template — though the paper does not exhibit one.

Limitations and open questions

The paper is candid about what remains undetermined. For odd prime powers $r$33 and $r$34, the exact value of $r$35 is open: no nontrivial maximal arc exists in $r$36, and it is unknown whether a non-projective intersecting block can still attain $r$37 or whether the truncated-plane value $r$38 is optimal in some cases. The stability question — whether pairs with $r$39 are forced into a geometric model after discarding lower-order incidences — is posed but unresolved. Whether the safe-block framework can beat the intersecting ceiling at all is likewise open, as is the optimal port cost $r$40 for intermediate shadow graphs, interpolating between the constant cost of empty shadows and the $r$41 scale of complete shadows at affine-plane orders. All asymptotic guarantees carry $r$42 error terms inherited from the affine-array row count, improving to $r$43 only when the port shadow is empty.

Conclusion

The paper reduces the problem of optimizing intersecting-block amalgamations for crown-free linear hypergraphs to a single parameter $r$44, bounds it universally by $r$45, characterizes its equality cases through pairwise balanced designs and maximal arcs, and attains it asymptotically for all linear-size crowns via truncated projective planes and padding. The safe-block extension frames the search for improvements beyond the intersecting ceiling as a concrete combinatorial question, while the rank obstruction shows that global arguments must respect the amalgamation mechanism underlying the entire construction.