--- title: Zarankiewicz Numbers in Extremal Graph Theory url: https://www.emergentmind.com/topics/zarankiewicz-numbers type: topic --- # Zarankiewicz Numbers in Extremal Graph Theory 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 [1303.5475]. The case \(s=t=2\), usually written \(z(m,n)\), is the \(C_4\)-free case and occupies a central position in extremal graph theory, incidence geometry, and several newer combinatorial frameworks tied to sum-of-squares rank [2410.03702]. ## 1. Classical definition and foundational bounds The classical Zarankiewicz number is defined by \[ z(m,n;s,t)=\max\{|E(G)|:\ G\subseteq K_{m,n},\ K_{s,t}\not\subseteq G\}, \] with the balanced notation \(z(n;t)\) or \(z(n,n;s,t)\) used in several sources when the two parts have equal size [1303.5475]. In the quadrilateral case one writes \(z(m,n)=z(m,n;2,2)\), and forbidding \(K_{2,2}\) is equivalent to forbidding \(C_4\) in bipartite graphs [2604.04111]. The basic upper-bound framework is the Kővári–Sós–Turán theorem. In one standard asymmetric form, \[ z(m,n;s,t)\le (t-1)^{1/s}\, m\, n^{1-1/s} + (s-1)\,n, \] with the symmetric counterpart obtained by swapping the parts and parameters [2007.12816]. In the \(C_4\)-free specialization, the standard bounds recorded in the recent literature include \[ z(m,n)\le n\sqrt m + m,\qquad z(m,n)\le m\sqrt n + n, \] and, in the symmetric case, \(z(n,n)=O(n^{3/2})\) [2603.04912]. For \(K_{2,2}\), more precise classical formulas include \[ z(m,n)\le \frac{m+\sqrt{m^2+4mn(n-1)}}{2}, \] and in the symmetric case \[ z(m)\le \frac{m+m\sqrt{4m-3}}{2}, \] with asymptotic normalization \[ \lim_{n\to\infty}\frac{z(n)}{n^{3/2}}=1 \] appearing in exact-work around projective-plane parameters [1303.5475]. 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 [2410.03702]. ## 2. The \(K_{2,2}\)-free case and exact small values The case \(z(m,n)=z(m,n;2,2)\) is the most extensively resolved classical regime. Finite projective planes provide the canonical exact constructions: if \(q\) is a prime power, then the point–line incidence graph of the projective plane of order \(q\) is \(C_4\)-free, \((q+1)\)-regular, and has \[ z(q^2+q+1)=(q+1)(q^2+q+1)=q^3+2q^2+2q+1 \] edges [1303.5475]. The same paper records exact asymmetric equality \[ z(p^2+p,p^2)=p^2(p+1) \] for primes \(p\) [1303.5475]. A substantial exact theory is known near the projective-plane sizes. Writing \(n=k^2+k+1-h\) with \(k\) a prime power, the exact values \[ z(n)= \begin{cases} k^3+2k^2+2k+1,& h=0,\\ k^3+2k^2,& h=1,\\ k^3+2k^2-2k,& h=2,\\ k^3+2k^2-4k+1,& h=3 \end{cases} \] are established, and the value for \(h=4\) is conjectured to be \(k^3+2k^2-6k+2\) [1303.5475]. Exact symmetric values are also listed for \(1\le n\le 21\), including \[ z(17)=74,\ z(18)=81,\ z(19)=88,\ z(20)=96,\ z(21)=105 \] [1303.5475]. Several small unbalanced values recur in later work on augmented variants. The exact values \[ z(m,2)=m+1,\quad z(3,3)=6,\quad z(4,3)=7,\quad z(4,4)=9,\quad z(5,3)=8,\quad z(5,4)=10,\quad z(5,5)=12,\quad z(6,5)=14 \] are explicitly cited in the limited augmented literature [2604.04111]. These values serve as the classical baseline for later quantities such as \(z_L(m,n)\), \(z_2(m,n)\), and \(z_{3L}(m,n)\), each of which preserves the \(C_4\)-free core while allowing controlled higher-order edge objects [2604.04111]. A related exact line of work studies unbalanced \(Z_{2,t}(m,n)\) when \(n=\Theta(tm^2)\). In that regime, Roman’s piecewise-linear upper bounds and new supplementary constraints determine the exact value in almost all remaining cases for large \(m\), and the simple terminal formula \[ Z_{2,t}(m,n)=(t-1)\binom{m}{2}+n \] holds whenever \(n\ge (t-1)\binom{m}{2}\) [2202.05507]. ## 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 \(n_i\) counting edges of size \(i\), objective \(\sum i n_i\), and constraints \[ \sum n_i=n,\qquad \sum \binom{i}{s}n_i \le (t-1)\binom{m}{s} \] [2411.18842]. This perspective has recently been sharpened by adding a new family of valid inequalities indexed by \(v\) and \(k\), yielding improved upper bounds on many small parameter sets and a new closed-form family of bounds \(B_k(m,n;s,t)\) that generalizes the \(s=2\) result of Chen–Horsley–Mammoliti [2411.18842]. Supersaturation provides a complementary refinement. For \(F=C_4\), the function \(F(n+n,m)\) denotes the minimum number of copies of \(F\) among all subgraphs of \(K_{n,n}\) with \(m\) edges. In the \(C_4\) case, one has the codegree identity \[ N_{C_4}(G)=\sum_{\{a,a'\}\subseteq A}\binom{\operatorname{codeg}(a,a')}{2}, \] and Jensen-type inequalities yield explicit lower bounds on \(C_4(n+n,m)\) once \(m\) exceeds the Zarankiewicz threshold [1711.09282]. In a sharp finite-geometric regime, if \(n=q^2+q+1\) and \(m\le z(n,n;C_4)+n\), the discrete Jensen lower bound is attained exactly by augmenting the projective-plane incidence graph with a suitable matching [1711.09282]. 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 \(z(b;s)\) for \(3\le s\le 6\), leading in particular to the exact bipartite Ramsey value \(b(2,5)=17\) and bounds \(17\le b(2,2,3)\le 18\) [1604.01257]. More recently, reinforced LLM-guided evolutionary search produced exact values \[ Z(11,21,3,3)=116,\quad Z(11,22,3,3)=121,\quad Z(12,22,3,3)=132, \] together with 41 further lower bounds for open \(K_{3,3}\)-free instances [2605.01120]. 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 \(r\)-uniform, \(r\)-partite analogue replaces \(K_{s,t}\) by an ordered complete \(r\)-partite \(r\)-graph \(K_{s_1,\dots,s_r}\). The corresponding quantity \[ z(m_1,\dots,m_r;s_1,\dots,s_r) \] is the maximum number of edges in an \(r\)-partite \(r\)-graph with part sizes \(m_i\) that contains no ordered \(K_{s_1,\dots,s_r}\) [2510.14869]. A hypergraph KST-type upper bound of the form \[ z(m_1,\dots,m_r;s_1,\dots,s_r)=O\!\Big(m_1\cdots m_{r-1}\, m_r^{\,1-\frac1{s_1\cdots s_{r-1}}}\Big) \] is derived from a supersaturation theorem, and a matching lower bound is proved in a broad parameter range by a random algebraic method [2510.14869]. Under the conditions \(s:=\prod_{i=1}^{r-1}s_i\le t\) and \(m:=\prod_{i=1}^{r-1}m_i \le n^{\,t^{1/(s-1)}/(s(s-1))}\), one obtains \[ z(m_1,\dots,m_{r-1},n;s_1,\dots,s_{r-1},t) =\Theta\!\Big(m_1\cdots m_{r-1}\, n^{\,1-\frac1{s_1\cdots s_{r-1}}}\Big) \] [2510.14869]. A distinct hypergraph line studies \(Z_{2,2}(m,n)\) through linear hypergraphs. Here \(Z_{2,2}(m,n)\) is the maximum total degree of a linear hypergraph with \(m\) vertices and \(n\) edges, equivalently the classical \(K_{2,2}\)-free matrix problem. Near the Steiner triple-system threshold \(n\approx \binom{m}{2}/3\), the exact value is determined by three linear expressions \[ U^+=\frac14 m(m-1)+\frac32 n,\qquad U^0=\frac1{14}m(3m-4)+\frac{12}{7}n,\qquad U^-=\frac16 m(m-1)+2n, \] with precise piecewise formulas above the triple threshold for all \(m\), and below it for sufficiently large \(m\) down to \(n\ge \binom{m}{2}/6 + m/3 + 40\) [2310.12685]. This threshold analysis shows that the \(K_{2,2}\)-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 \(n^{2-1/s}\) scale to near-linear behavior [2410.03702]. For axis-parallel boxes with prescribed direction sets, a sharp dichotomy has been proved: the geometric Zarankiewicz number \(z_d^{\vec F}(n;t)\) is either \[ \Theta_r(tn^{r-1}) \] in the non-\(2\)-coherent case, or at least \[ \Omega\!\Big(tn^{r-1}\frac{\log n}{\log\log n}\Big) \] in the \(2\)-coherent case [2604.20815]. Here \(2\)-coherence means that all but at most one family share at least two common varying coordinates, thereby permitting a hidden planar incidence structure [2604.20815]. Related sharp linear bounds are known for several low-dimensional geometric graph classes. For bipartite graphs of Ferrers dimension three, \[ Z_{\mathrm{CHAIN}^3}(n;k)\le 9n(k-1), \] whereas Ferrers dimension four already permits \[ \Omega\!\Big(nk\frac{\log n}{\log\log n}\Big) \] edges in \(K_{k,k}\)-free examples [2510.20737]. In the same work, chordal bipartite graphs satisfy the tight linear bound \[ Z_{\mathrm{Chordal}}(n;k)\le 2n(k-1), \] and grid intersection graphs satisfy \[ Z_{\mathrm{GIG}}(n;k)\le 54n(k-1) \] [2510.20737]. 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 \(O(|V|^{r-1})\) bounds on definable \(r\)-hypergraphs under \(K_{k,\dots,k}\)-freeness. Such bounds are established for semibounded o-minimal structures on sufficiently distant grids, Presburger arithmetic, \(\langle \mathbb R,<,+,\mathbb Z\rangle\), stable \(1\)-based structures without the finite cover property, and locally modular regular types [2510.03546]. 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 [2510.03546]. 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 \(C_4\)-free extremal configurations as lower bounds for the sum-of-squares rank of biquadratic forms. For an \(m\times n\) SOS biquadratic form \(P\), the maximum possible SOS rank is denoted \(\operatorname{BSR}(m,n)\). Classical \(C_4\)-free graphs yield simple irreducible forms and therefore \[ \operatorname{BSR}(m,n)\ge z(m,n) \] [2603.04912]. The first augmentation introduces \(2\)-edges, where a pair \((i,j;k,l)\) represents the square \[ (x_i y_j + x_k y_l)^2. \] The resulting double Zarankiewicz number \(z_2(m,n)\) is defined as the maximum \(|E_1|+|E_2|\) over bipartite configurations with ordinary edges and \(2\)-edges that contain no generalized \(C_4\)-cycle [2603.04912]. Exact values include \[ z_2(m,2)=m+1,\quad z_2(3,3)=6,\quad z_2(4,3)=8,\quad z_2(5,3)=9, \] and \[ 10\le z_2(4,4)\le 11 \] [2603.04912]. In particular, \[ z_2(4,3)=8>z(4,3)=7,\qquad z_2(5,3)=9>z(5,3)=8, \] showing that the classical Zarankiewicz number can underestimate SOS rank lower bounds [2603.04912]. The limited augmented number \(z_L(m,n)\) fixes the \(1\)-edge part to have size \(z(m,n)\) and adds admissible \(2\)-edges. It satisfies \[ \operatorname{BSR}(m,n)\ge z_L(m,n)\ge z(m,n), \] and exact values are now known for all \(m,n\le 5\), including \[ z_L(5,3)=9,\qquad z_L(5,4)=12,\qquad z_L(5,5)=14, \] together with \[ z_L(6,5)\ge 17 \] [2604.04111]. In the incidence-graph family of complete graphs, computational work gives \[ z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88 \] [2605.29658]. A further extension introduces \(3\)-edges \((i,j;k,l;p,q)\), representing \[ (x_i y_j + x_k y_l + x_p y_q)^2. \] The associated quantities \(z_{3L}(m,n)\) and \(z_{3A}(m,n)\) satisfy \[ z_{3A}(m,n)\ge z_{3L}(m,n)\ge z_L(m,n)\ge z(m,n), \] and the main theorem states that for a generalized cycle-free simple \(3\)-edge-augmented graph, \[ \operatorname{sos}(P_G)=|E_1|+|E_2|+|E_3| \] [2605.09926]. Concrete improvements include \[ z_{3L}(5,3)=10,\qquad z_{3L}(6,4)\ge 16,\qquad z_{3L}(5,5)\ge 16, \] which improve the corresponding \(2\)-edge bounds \(z_L(5,3)=9\), \(z_L(6,4)=14\), and \(z_L(5,5)=14\) [2605.09926]. These augmented parameters are not classical Zarankiewicz numbers in the usual extremal-graph sense. They are, however, explicit extensions of the \(C_4\)-free paradigm, and they formalize a new combinatorial principle: forbidding generalized \(4\)-cycle obstructions can force additivity of bilinear squares in SOS decompositions. That principle has already produced sharper lower bounds than the classical \(z(m,n)\) in several small dimensions [2605.09926]. ## 7. Relations to Ramsey theory and current directions Zarankiewicz numbers are closely tied to bipartite Ramsey numbers. In the \(C_4\)-avoidance setting, if every color class in a \(k\)-coloring of \(K_{n,n}\) is \(C_4\)-free, then each color class has at most \(z(n)\) edges, so \(kz(n)