Cross Intersecting Systems in Extremal Combinatorics

• Cross intersecting systems are fundamental structures in extremal combinatorics defined by families of disjoint set pairs with prescribed intersection sizes.
• Classical results like Bollobás’s theorem provide sharp upper bounds using weight-sum arguments, establishing key limits for these systems.
• Recent advances refine these bounds through induction, case analysis, and combinatorial inequalities, exemplified by the unique 5-cycle extremal construction.

A cross intersecting system is a fundamental structure in extremal combinatorics consisting of collections of subsets (typically pairs or families) defined over a ground set, subject to constraints on the intersection sizes between selected pairs. The prototypical objects are set-pair systems of ordered pairs $(A_i, B_i)$, and generalizations to higher arities or intersection multiplicities are central in the contemporary theory. A sharp line of investigation focuses on systems where every off-diagonal intersection has prescribed exact cardinality—especially the 1-cross intersecting set-pair systems, as detailed by Holzman (Holzman, 2020).

1. Formal Framework and Definitions

Let $V$ be a finite ground set. A set-pair system (SPS) is a family

$$S = \{ (A_i, B_i) \}_{i=1}^m, \quad A_i, B_i \subseteq V, \ A_i \cap B_i = \emptyset.$$

The system is cross-intersecting if $\forall i \ne j : A_i \cap B_j \ne \emptyset$. It is 1-cross intersecting if, additionally, for all $i \ne j$, $|A_i \cap B_j| = 1$. Typically, one imposes size bounds $|A_i| \leq a$, $|B_i| \leq b$ for all $i$, and denotes the maximal system size by $m = m(a, b)$.

For general $S = \{(A_i, B_i)\}_{i \in I}$, the ground set and a key potential function are:

• $V(S) = \bigcup_{i \in I} (A_i \cup B_i)$,
• $$\Sigma(S) = \sum_{i \in I} 1/\binom{|A_i|+|B_i|}{|A_i|}$$.

This formalism is designed to handle sharp extremal bounds and reductions pertinent to intersection properties.

2. Classical Result: Bollobás’s Set-Pair Theorem

Bollobás (1965) established that for any cross-intersecting set-pair system with $|A_i| \leq a$, $|B_i| \leq b$, the maximum number of pairs satisfies

$m \leq \binom{a+b}{a}.$

This follows from a weight-sum argument: each pair is assigned a weight $1/\binom{|A_i|+|B_i|}{|A_i|}$, and

$$\displaystyle \sum_{i=1}^m 1/\binom{|A_i|+|B_i|}{|A_i|} \leq 1;$$

in the uniform case, $\Sigma(S) = m/\binom{a+b}{a}$, recovering the stated bound.

3. 1-Cross Intersecting Systems: New Upper Bounds

The Füredi–Gyárfás–Király conjecture proposed that 1-cross constraints admit a strict reduction below the Bollobás threshold, uniform for all $n$. Holzman confirmed this, proving:

Let $S = \{(A_i, B_i)\}_{i=1}^m$ be 1-cross intersecting with $|A_i| \leq a, |B_i| \leq b, \ a, b \geq 2$. Then

$$m \leq \frac{29}{30} \binom{a+b}{a} \qquad \text{and} \qquad \Sigma(S) \leq 29/30.$$

The proof employs:

• a reduction lemma eliminating non-intersecting elements;
• induction on $|V(S)|$ using the identity

$$\Sigma(S) = \frac1{|V(S)|}\sum_{v\in V(S)}\Sigma(S[I^A_{\bar v}]-\{v\});$$

• tight binomial ratio bounds, ensuring weight contributions strictly below 1 unless in small exceptional cases.

4. Recent Improvement and Extremality

Kostochka–McCourtt–Nahvi strengthened Holzman's bound, verifying his conjecture that

$m \leq \frac{5}{6} \binom{a+b}{a}$

whenever $a,b \geq 2$, and that the weighted sum $\Sigma(S) \leq 5/6$ exactly, matching the known extremal construction for $(a, b) = (2, 2)$. The unique extremal example is the "5-cycle" on $\mathbb{Z}/5\mathbb{Z}$: $$A_i = \{i, i+1\},\ B_i = \{i-1, i+2\},\ i = 0, ..., 4.$$ Their approach refines the inductive and combinatorial weight arguments and performs detailed casework on small pairs, accompanied by sharp inequalities for binomial coefficients (Kostochka et al., 2021).

5. Sharpness, Constructions, and Limitations

• For $(a, b) = (2, 2)$:

$m_{\max} = 5 = \frac{5}{6} \binom{4}{2}.$

• For large $n$ in the symmetric case, Füredi–Gyárfás–Király showed

$m(n, n, 1) \geq 5^{n/2}$

via tensor products of small extremal blocks.

• The best current constructions do not rule out the existence of an exponential upper bound $C^n$ with $C < 4$, and the tight asymptotics for large $a, b$ remain open.

A summary of bounds appears below:

Regime	Best Lower Bound	Best Upper Bound	Sharp Example
(a, b) = (2, 2)	5	5	5-cycle
n even, a = b = n	$5^{n/2}$	$(5/6)\binom{2n}{n}$	Tensor powers
a, b ≥ 2	(by construction)	$(5/6)\binom{a+b}{a}$	5-cycle when a=b=2

6. Proof Techniques and Methodology

The recent advances rest on:

• Weight-sum induction and vertex-removal arguments (analogous to Bollobás’s setting but more delicate due to exact intersection numbers) (Holzman, 2020).
• Bounding small configurations (notably, contributions from (2, 2)-pairs) by direct calculation and combinatorial inequalities.
• Exploiting symmetry and minimality within the system to partition into types and draw global conclusions.

7. Open Problems and Directions

• Is the extremal multiplicative constant $5/6$ optimal for all sufficiently large $a, b$?
• Does there exist a sequence $\epsilon_n \to 0$ so that $m(n, n, 1) \leq \epsilon_n \binom{2n}{n}$?
• What is the precise exponential rate for large uniform systems in the 1-cross regime?
• How do analogous bounds behave for $t$-cross intersection (i.e., $|A_i \cap B_j| = t$ for $i \ne j$), and which combinatorial or algebraic techniques will extend?

The additional rigidity imposed by 1-cross intersection constraints yields new combinatorial phenomena strictly improving classical extremal bounds, and advances understanding in the intersection theory of combinatorial structures (Holzman, 2020, Kostochka et al., 2021).