Set systems containing no singleton intersection and the Delsarte number

Abstract: We prove that the maximum size of a family of $k$-element subsets of the set $[n] = {1, 2, \ldots, n}$ which contains no singleton intersection is $\binom{n-2}{k-2}$ when $3k-3 \le n \le k<sup>2-k+1$. This improves upon a recent result of Cherkashin. Our proof uses Schrijver's variant of the Lovász number and furnishes an infinite family of graphs where the Schrijver variant of the Lovász number is strictly smaller than the Lovász number. As a consequence of our result and a recent result of Keller and Lifshitz, it follows that for $k$ sufficiently large, the maximum size of a $k$-uniform family on $[n]$ containing no singleton intersection is $\binom{n-2}{k-2}$ for all $n\ge 3k-3$, which is the best possible threshold.

• The paper establishes that any k-uniform family with no singleton intersections has maximum size exactly binom(n-2, k-2) for 3k-3 ≤ n ≤ k^2-k+1.
• It employs the Delsarte linear programming method and Wilson’s eigenvalue techniques to derive sharp bounds that align with classical Erdős–Ko–Rado thresholds.
• The work identifies an infinite graph family where Schrijver’s Delsarte number is strictly smaller than the Lovász number, revealing novel structural insights.

Set Systems Avoiding Singleton Intersection and the Delsarte Number

Summary of Main Results

The paper provides a significant advancement in extremal set theory by determining the precise maximum size of a $k$-uniform family of subsets of $[n]$ with no intersection of size exactly one, for the parameter regime $3k-3 \leq n \leq k^2-k+1$. The result asserts that for such $n$ and $k \geq 3$, the largest possible family $F \subset \binom{[n]}{k}$ with no pair $F, F'$ satisfying $|F \cap F'| = 1$ has size exactly $\binom{n-2}{k-2}$. This bound is shown to be tight, matching the classical threshold for $2$-intersecting families. The proof leverages algebraic bounds from the Delsarte linear programming method, realized via Schrijver’s variant of the Lovász number, and employs Wilson's approach to eigenvalue computations of associated Johnson-type graphs.

Additionally, the work exhibits an infinite family of graphs where Schrijver's variant of the Lovász number (the Delsarte number, $[n]$0) is strictly smaller than the classical Lovász number ($[n]$1), a rare and structurally informative phenomenon. Corollaries and generalizations covering related set systems are also derived using these spectral techniques.

Context and Prior Work

The classical Erdős-Ko-Rado (EKR) theorem characterizes the size and structure of intersecting families in extremal combinatorics. The forbidden intersection problem—identifying the maximal size of a family omitting a specified intersection size—generalizes this framework. For singleton-free intersection sets, Frankl’s theorem established for large $[n]$2 that the extremal size is $[n]$3, but the minimal threshold and exact results for smaller values of $[n]$4 remained unresolved.

Cherkashin recently used the Hoffman bound to achieve a threshold of $[n]$5 for the singleton-free regime, yielding size at most $[n]$6. The current work supersedes this by closing the gap down to $[n]$7, which is shown to be the optimal lower bound by referencing known constructions.

Recent advances by Keller and Lifshitz, and related stability results, further supply general frameworks for Turán-type problems on hypergraphs, contextualizing the maximal families obtained as $[n]$8-stars and confirming the asymptotic tightness of the bounds for large $[n]$9.

Technical Approach

The method centers on analyzing the independence number of the generalized Johnson graph $3k-3 \leq n \leq k^2-k+1$0, where vertices are $3k-3 \leq n \leq k^2-k+1$1-sets and adjacency is defined by forbidden intersection sizes. The extremal problem for $3k-3 \leq n \leq k^2-k+1$2-systems maps directly to $3k-3 \leq n \leq k^2-k+1$3.

Central to the argument is the application of Schrijver’s $3k-3 \leq n \leq k^2-k+1$4 (Delsarte number). The author constructs a matrix $3k-3 \leq n \leq k^2-k+1$5, where $3k-3 \leq n \leq k^2-k+1$6 is the all-ones matrix and $3k-3 \leq n \leq k^2-k+1$7 is a linear combination of matrices indexed by intersection cardinalities—Wilson’s matrix, tailored for the Erdős-Ko-Rado setting at $3k-3 \leq n \leq k^2-k+1$8. Wilson’s previous work provides sharp eigenvalue bounds, here adapted to certify that $3k-3 \leq n \leq k^2-k+1$9 dominates the constraints in Schrijver’s program and has maximal eigenvalue $n$0, precisely matching the claimed independence bound.

For $n$1, the transition point, $n$2 simultaneously affirms the Lovász bound, so $n$3 there. For all $n$4, the analysis shows that the classical Lovász number strictly exceeds the Delsarte bound, underlining a structural difference between the two.

Explicit Numerical and Theoretical Claims

• For all $n$5 and $n$6, any family $n$7 with no singleton intersections satisfies $n$8.
• When $n$9, this upper bound coincides with the Lovász number.
• For $k \geq 3$0, Schrijver's Delsarte number is provably strictly less than the Lovász number for $k \geq 3$1, exhibiting an infinite family with this separation.
• For $k \geq 3$2, there exist $k \geq 3$3-intersecting families exceeding the $k \geq 3$4 size, affirming the sharpness of the threshold.
• For large $k \geq 3$5, both the Lovász and Delsarte numbers grow as $k \geq 3$6, which vastly exceeds the actual independence number, so spectral bounds alone are insufficient in this asymptotic regime.

Implications and Future Directions

From a combinatorial standpoint, this result tightens the quantitative understanding of forbidden intersection set systems, precisely delineating the extremal function for all $k \geq 3$7, which was previously uncharacterized for much of this range. The explicit display of families of graphs with $k \geq 3$8 contributes to the study of semidefinite relaxations in algebraic combinatorics and motivates deeper exploration into integrality gaps and hierarchies of spectral bounds within association schemes.

Practically, the techniques reinforce the utility of analytic and algebraic combinatorics in hypergraph extremal problems. The methods for constructing feasible matrices for Delsarte’s program using association scheme idempotents and structure constants could extend to broader forbidden intersection configurations and suggest analogous improvements in related Turán-type problems.

Possible future research directions inspired by this work include:

• Classifying all parameter regimes where Schrijver's and Lovász's theta functions diverge for other families of combinatorial graphs.
• Extending sharp forbidden intersection results to wider classes of intersection configurations $k \geq 3$9, especially those with more than one omitted intersection size.
• Exploring the stability and structural characterization of nearly-extremal families under these constraints.

Conclusion

The paper settles the extremal problem for $F \subset \binom{[n]}{k}$0-uniform set families with no intersection of size one for a broad, previously inaccessible range of $F \subset \binom{[n]}{k}$1, via spectral methods rooted in association schemes and semidefinite programming. The work also elucidates the relationship between the Delsarte and Lovász numbers, providing new infinite graph families exhibiting strict separation. These results both advance the theory of extremal set systems and enrich the algebraic combinatorics toolkit for related intersection problems.