Zarankiewicz Numbers in Extremal Graph Theory

• Zarankiewicz numbers are extremal parameters that determine the maximum edge density in bipartite graphs (or 0–1 matrices) without prescribed complete bipartite subgraphs like K₍s,t₎.
• They are analyzed through methods such as combinatorial counting, linear programming, and finite geometry, with classical bounds provided by the Kővári–Sós–Turán theorem.
• Recent advances extend these numbers to hypergraph settings and augmented versions, impacting applications in Ramsey theory and sum-of-squares decompositions.

Zarankiewicz numbers are extremal parameters that measure how dense a bipartite graph, or equivalently a $0$–$1$ matrix, can be while avoiding a prescribed complete bipartite subgraph. In the classical formulation, for integers $m,n,s,t \ge 1$, the number $z(m,n;s,t)$ is the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that contains no $K_{s,t}$; in matrix language, it is the maximum number of $1$-entries in an $m\times n$ matrix with no $s\times t$ all-ones submatrix (Dybizbański et al., 2013). The case $1$0, usually written $1$1, is the $1$2-free case and occupies a central position in extremal graph theory, incidence geometry, and several newer combinatorial frameworks tied to sum-of-squares rank (Smorodinsky, 2024).

1. Classical definition and foundational bounds

The classical Zarankiewicz number is defined by

$1$3

with the balanced notation $1$4 or $1$5 used in several sources when the two parts have equal size (Dybizbański et al., 2013). In the quadrilateral case one writes $1$6, and forbidding $1$7 is equivalent to forbidding $1$8 in bipartite graphs (Qi et al., 5 Apr 2026).

The basic upper-bound framework is the Kővári–Sós–Turán theorem. In one standard asymmetric form,

$1$9

with the symmetric counterpart obtained by swapping the parts and parameters (Conlon, 2020). In the $m,n,s,t \ge 1$0-free specialization, the standard bounds recorded in the recent literature include

$m,n,s,t \ge 1$1

and, in the symmetric case, $m,n,s,t \ge 1$2 (Qi et al., 5 Mar 2026). For $m,n,s,t \ge 1$3, more precise classical formulas include

$m,n,s,t \ge 1$4

and in the symmetric case

$m,n,s,t \ge 1$5

with asymptotic normalization

$m,n,s,t \ge 1$6

appearing in exact-work around projective-plane parameters (Dybizbański et al., 2013).

These inequalities fix the exponent but generally not the exact value. The problem is therefore simultaneously asymptotic and arithmetic: the Kővári–Sós–Turán theorem determines the correct order in many regimes, while exact values often depend on finite-geometric constructions or delicate counting arguments (Smorodinsky, 2024).

2. The $m,n,s,t \ge 1$7-free case and exact small values

The case $m,n,s,t \ge 1$8 is the most extensively resolved classical regime. Finite projective planes provide the canonical exact constructions: if $m,n,s,t \ge 1$9 is a prime power, then the point–line incidence graph of the projective plane of order $z(m,n;s,t)$0 is $z(m,n;s,t)$1-free, $z(m,n;s,t)$2-regular, and has

$z(m,n;s,t)$3

edges (Dybizbański et al., 2013). The same paper records exact asymmetric equality

$z(m,n;s,t)$4

for primes $z(m,n;s,t)$5 (Dybizbański et al., 2013).

A substantial exact theory is known near the projective-plane sizes. Writing $z(m,n;s,t)$6 with $z(m,n;s,t)$7 a prime power, the exact values

$z(m,n;s,t)$8

are established, and the value for $z(m,n;s,t)$9 is conjectured to be $m$0 (Dybizbański et al., 2013). Exact symmetric values are also listed for $m$1, including

$m$2

(Dybizbański et al., 2013).

Several small unbalanced values recur in later work on augmented variants. The exact values

$m$3

are explicitly cited in the limited augmented literature (Qi et al., 5 Apr 2026). These values serve as the classical baseline for later quantities such as $m$4, $m$5, and $m$6, each of which preserves the $m$7-free core while allowing controlled higher-order edge objects (Qi et al., 5 Apr 2026).

A related exact line of work studies unbalanced $m$8 when $m$9. In that regime, Roman’s piecewise-linear upper bounds and new supplementary constraints determine the exact value in almost all remaining cases for large $n$0, and the simple terminal formula

$n$1

holds whenever $n$2 (Chen et al., 2022).

3. Structural methods: linear programming, supersaturation, and computation

A major modern viewpoint treats Zarankiewicz numbers through optimization over incidence vectors. Roman’s bound can be expressed as the optimum of a linear program in the hypergraph formulation, with variables $n$3 counting edges of size $n$4, objective $n$5, and constraints

$n$6

(Davies et al., 2024). This perspective has recently been sharpened by adding a new family of valid inequalities indexed by $n$7 and $n$8, yielding improved upper bounds on many small parameter sets and a new closed-form family of bounds $n$9 that generalizes the $K_{s,t}$0 result of Chen–Horsley–Mammoliti (Davies et al., 2024).

Supersaturation provides a complementary refinement. For $K_{s,t}$1, the function $K_{s,t}$2 denotes the minimum number of copies of $K_{s,t}$3 among all subgraphs of $K_{s,t}$4 with $K_{s,t}$5 edges. In the $K_{s,t}$6 case, one has the codegree identity

$K_{s,t}$7

and Jensen-type inequalities yield explicit lower bounds on $K_{s,t}$8 once $K_{s,t}$9 exceeds the Zarankiewicz threshold (Nagy, 2017). In a sharp finite-geometric regime, if $1$0 and $1$1, the discrete Jensen lower bound is attained exactly by augmenting the projective-plane incidence graph with a suitable matching (Nagy, 2017). This places the classical threshold into a supersaturated continuum linking extremal constructions to random-graph asymptotics.

Computation has become increasingly important. SAT/IP techniques have been used to determine or bound many balanced Zarankiewicz numbers $1$2 for $1$3, leading in particular to the exact bipartite Ramsey value $1$4 and bounds $1$5 (Collins et al., 2016). More recently, reinforced LLM-guided evolutionary search produced exact values

$1$6

together with 41 further lower bounds for open $1$7-free instances (Bhan et al., 1 May 2026). This suggests that constructive extremal search is becoming a meaningful complement to classical LP and finite-geometry approaches.

4. Hypergraph generalizations and threshold phenomena

The direct $1$8-uniform, $1$9-partite analogue replaces $m\times n$0 by an ordered complete $m\times n$1-partite $m\times n$2-graph $m\times n$3. The corresponding quantity

$m\times n$4

is the maximum number of edges in an $m\times n$5-partite $m\times n$6-graph with part sizes $m\times n$7 that contains no ordered $m\times n$8 (Gao et al., 16 Oct 2025). A hypergraph KST-type upper bound of the form

$m\times n$9

is derived from a supersaturation theorem, and a matching lower bound is proved in a broad parameter range by a random algebraic method (Gao et al., 16 Oct 2025). Under the conditions $s\times t$0 and $s\times t$1, one obtains

$s\times t$2

(Gao et al., 16 Oct 2025).

A distinct hypergraph line studies $s\times t$3 through linear hypergraphs. Here $s\times t$4 is the maximum total degree of a linear hypergraph with $s\times t$5 vertices and $s\times t$6 edges, equivalently the classical $s\times t$7-free matrix problem. Near the Steiner triple-system threshold $s\times t$8, the exact value is determined by three linear expressions

$s\times t$9

with precise piecewise formulas above the triple threshold for all $1$00, and below it for sufficiently large $1$01 down to $1$02 (Chen et al., 2023). This threshold analysis shows that the $1$03-free problem is not merely asymptotic: the exact regime changes according to the edge-size profile of the corresponding linear hypergraph.

A plausible implication is that Zarankiewicz theory in hypergraphs is governed by a mixture of KST exponents, design-theoretic divisibility, and algebraic geometry. That synthesis is explicit in the current literature, rather than implicit.

5. Geometric and model-theoretic Zarankiewicz problems

A large body of recent work concerns Zarankiewicz problems inside structured graph classes. In geometry, the guiding question is when geometric constraints force the extremal growth to collapse from the classical $1$04 scale to near-linear behavior (Smorodinsky, 2024). For axis-parallel boxes with prescribed direction sets, a sharp dichotomy has been proved: the geometric Zarankiewicz number $1$05 is either

$1$06

in the non-$1$07-coherent case, or at least

$1$08

in the $1$09-coherent case (Chao et al., 22 Apr 2026). Here $1$10-coherence means that all but at most one family share at least two common varying coordinates, thereby permitting a hidden planar incidence structure (Chao et al., 22 Apr 2026).

Related sharp linear bounds are known for several low-dimensional geometric graph classes. For bipartite graphs of Ferrers dimension three,

$1$11

whereas Ferrers dimension four already permits

$1$12

edges in $1$13-free examples (Chalermsook et al., 23 Oct 2025). In the same work, chordal bipartite graphs satisfy the tight linear bound

$1$14

and grid intersection graphs satisfy

$1$15

(Chalermsook et al., 23 Oct 2025). These results show that the phrase “Zarankiewicz number” now frequently denotes a class-restricted extremal function, not only the unrestricted bipartite one.

The model-theoretic “global linear Zarankiewicz problem” asks for linear $1$16 bounds on definable $1$17-hypergraphs under $1$18-freeness. Such bounds are established for semibounded o-minimal structures on sufficiently distant grids, Presburger arithmetic, $1$19, stable $1$20-based structures without the finite cover property, and locally modular regular types (Eleftheriou et al., 3 Oct 2025). In the binary case, one consequence is a characterization: for reducts of a real closed field, non-definability of multiplication is equivalent to the existence of linear bounds on sufficiently distant definable incidences (Eleftheriou et al., 3 Oct 2025). This suggests that “linear Zarankiewicz behavior” can serve as a structural dividing line between tame additive geometries and field-like incidence geometries.

6. Augmented Zarankiewicz numbers and SOS rank

A notable recent development reinterprets $1$21-free extremal configurations as lower bounds for the sum-of-squares rank of biquadratic forms. For an $1$22 SOS biquadratic form $1$23, the maximum possible SOS rank is denoted $1$24. Classical $1$25-free graphs yield simple irreducible forms and therefore

$1$26

(Qi et al., 5 Mar 2026).

The first augmentation introduces $1$27-edges, where a pair $1$28 represents the square

$1$29

The resulting double Zarankiewicz number $1$30 is defined as the maximum $1$31 over bipartite configurations with ordinary edges and $1$32-edges that contain no generalized $1$33-cycle (Qi et al., 5 Mar 2026). Exact values include

$1$34

and

$1$35

(Qi et al., 5 Mar 2026). In particular,

$1$36

showing that the classical Zarankiewicz number can underestimate SOS rank lower bounds (Qi et al., 5 Mar 2026).

The limited augmented number $1$37 fixes the $1$38-edge part to have size $1$39 and adds admissible $1$40-edges. It satisfies

$1$41

and exact values are now known for all $1$42, including

$1$43

together with

$1$44

(Qi et al., 5 Apr 2026). In the incidence-graph family of complete graphs, computational work gives

$1$45

(Yi et al., 28 May 2026).

A further extension introduces $1$46-edges $1$47, representing

$1$48

The associated quantities $1$49 and $1$50 satisfy

$1$51

and the main theorem states that for a generalized cycle-free simple $1$52-edge-augmented graph,

$1$53

(Qi et al., 11 May 2026). Concrete improvements include

$1$54

which improve the corresponding $1$55-edge bounds $1$56, $1$57, and $1$58 (Qi et al., 11 May 2026).

These augmented parameters are not classical Zarankiewicz numbers in the usual extremal-graph sense. They are, however, explicit extensions of the $1$59-free paradigm, and they formalize a new combinatorial principle: forbidding generalized $1$60-cycle obstructions can force additivity of bilinear squares in SOS decompositions. That principle has already produced sharper lower bounds than the classical $1$61 in several small dimensions (Qi et al., 11 May 2026).

7. Relations to Ramsey theory and current directions

Zarankiewicz numbers are closely tied to bipartite Ramsey numbers. In the $1$62-avoidance setting, if every color class in a $1$63-coloring of $1$64 is $1$65-free, then each color class has at most $1$66 edges, so $1$67 implies a monochromatic $1$68 (Dybizbański et al., 2013). This mechanism yields, for example,

$1$69

using the exact values of $1$70 near projective-plane sizes (Dybizbański et al., 2013). More generally, improved tables of $1$71 feed directly into exact or near-exact bipartite Ramsey results such as $1$72 (Collins et al., 2016).

A different Ramsey connection uses a variant $1$73 derived from a graph $1$74, where $1$75 is a family of forbidden bipartite incidence patterns. Dense $1$76-free constructions imply lower bounds on $1$77, leading to

$1$78

(Conlon et al., 2023). This extends Zarankiewicz theory beyond complete bipartite forbidden subgraphs into Ramsey-oriented matrix avoidance.

Several broad open directions recur across the literature. Determining exact constants in

$1$79

remains open outside a few cases such as $1$80 and $1$81 (Smorodinsky, 2024). In the geometric setting, the exact order for general $1$82-coherent direction vectors is not known (Chao et al., 22 Apr 2026). In bounded VC-dimension, the Fox–Pach–Sheffer–Suk–Zahl bound $1$83 is known to be non-sharp for $1$84, since the true maximum is

$1$85

for $1$86-free bipartite graphs of VC-dimension at most $1$87 (Janzer et al., 2020). In the augmented SOS setting, tight values of $1$88 and $1$89, as well as higher $1$90-edge extensions, remain open (Qi et al., 11 May 2026).

Taken together, these directions indicate that Zarankiewicz numbers now name a family of extremal problems rather than a single function. The classical bipartite quantity remains the prototype, but current research treats it as a hub linking finite geometry, incidence theory, hypergraph extremal theory, bounded-complexity graph classes, model theory, and algebraic complexity.