Komlós Conjecture

• The Komlós Conjecture is a central problem in discrepancy theory, asserting the existence of a universal constant such that, for every collection of vectors v_i, one can choose signs such that the L_infinity norm of the signed sum is bounded by this constant.
• The conjecture generalizes the Beck-Fiala discrepancy problem and is related to vector-balancing, with significant implications for dynamic programming algorithms and integer linear programming.
• The Komlós Conjecture is distinct from the Loebl-Komlos-Sos tree containment conjecture and addresses bounded Euclidean norms and combinatorial discrepancies of real matrices.

The Komlós conjecture is a central problem in discrepancy theory concerning the signing of vectors with bounded Euclidean norm. It asserts that there exists a universal constant $C$ such that, for every collection $v_1,\ldots,v_n\in\mathbb R^d$ satisfying $\|v_i\|_2\le 1$, one can choose signs $\varepsilon_i\in\{-1,1\}$ with

$$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$

Equivalently, every real matrix whose columns have Euclidean norm at most one has discrepancy bounded by an absolute constant. The conjecture is distinct from the Loebl–Komlós–Sós tree-containment conjecture, although both belong to the broader family of problems associated with János Komlós.

1. Formulation and terminology

For a matrix $A\in\mathbb R^{m\times n}$, its combinatorial discrepancy is

$$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$

The matrix is in the Komlós setting when every column satisfies

$\|A_{*j}\|_2\le 1.$

The Komlós conjecture states that

$\operatorname{disc}(A)=O(1),$

with an implied constant independent of $m$, $v_1,\ldots,v_n\in\mathbb R^d$0, and $v_1,\ldots,v_n\in\mathbb R^d$1. In vector form, the conjecture asks for a signing whose coordinatewise signed sums remain uniformly bounded, regardless of the number of vectors or ambient dimension.

The normalization is invariant under the usual vector interpretation: the columns $v_1,\ldots,v_n\in\mathbb R^d$2 are the vectors $v_1,\ldots,v_n\in\mathbb R^d$3, and

$v_1,\ldots,v_n\in\mathbb R^d$4

The $v_1,\ldots,v_n\in\mathbb R^d$5 norm measures the largest coordinate of the resulting vector. The problem is therefore a vector-balancing problem in which the available coefficients are restricted to the two values $v_1,\ldots,v_n\in\mathbb R^d$6 and $v_1,\ldots,v_n\in\mathbb R^d$7.

The conjecture is stronger than results allowing fractional coefficients, partial colorings, arbitrary unit vectors as coefficients, or complex unit-modulus coefficients. In particular, a bound for complex discrepancy or rank-$v_1,\ldots,v_n\in\mathbb R^d$8 vector discrepancy does not imply the Boolean signing statement.

The term Komlós conjecture is also used in other areas of extremal graph theory. The Loebl–Komlós–Sós conjecture asserts that an $v_1,\ldots,v_n\in\mathbb R^d$9-vertex graph with at least $\|v_i\|_2\le 1$0 vertices of degree at least $\|v_i\|_2\le 1$1 contains every tree of order $\|v_i\|_2\le 1$2. It is a different conjecture, involving tree embeddings and degree distributions rather than vector discrepancy.

2. Relations with discrepancy theory

The Komlós problem generalizes the Beck–Fiala discrepancy problem. In a set system, the incidence matrix is a $\|v_i\|_2\le 1$3-$\|v_i\|_2\le 1$4 matrix. If every element belongs to at most $\|v_i\|_2\le 1$5 sets, each incidence-matrix column has Euclidean norm at most $\|v_i\|_2\le 1$6. Applying the Komlós conjecture to the matrix scaled by $\|v_i\|_2\le 1$7 would imply the Beck–Fiala bound

$\|v_i\|_2\le 1$8

Banaszczyk proved the general upper bound

$\|v_i\|_2\le 1$9

for matrices with unit-bounded column norms. This remains the principal classical benchmark in the general setting, although subsequent work has improved its dependence on $\varepsilon_i\in\{-1,1\}$0.

A partial-coloring theorem of Kashin gives a constant-discrepancy partial signing: a vector in $\varepsilon_i\in\{-1,1\}$1 with at least $\varepsilon_i\in\{-1,1\}$2 nonzero coordinates can be chosen while maintaining bounded discrepancy. Iterating this result gives a full-coloring bound of $\varepsilon_i\in\{-1,1\}$3, but the discrepancy cost accumulates over the successive partial-coloring stages. The gap between constant-discrepancy partial colorings and constant-discrepancy full colorings is one of the structural difficulties of the subject.

The discrepancy problem also admits several relaxations. In rank-$\varepsilon_i\in\{-1,1\}$4 vector discrepancy, the signs are replaced by unit vectors in $\varepsilon_i\in\{-1,1\}$5. In complex discrepancy, the coefficients are allowed to be arbitrary complex numbers of unit modulus. Gaussian discrepancy replaces the deterministic signing by a centered Gaussian vector with a constrained covariance matrix. These relaxations are generally easier than the Boolean problem.

3. Bounds before the resolution

The algorithmic result “An Algorithm for Komlós Conjecture Matching Banaszczyk’s Bound” gave an efficient randomized algorithm producing

$\varepsilon_i\in\{-1,1\}$6

discrepancy for Komlós matrices, matching Banaszczyk’s existential bound (Bansal et al., 2016). Its method used SDP-generated vector updates, random sign projections, freezing of nearly integral coordinates, approximate orthogonality, and Freedman-type martingale concentration.

A later algorithmic approach based on affine spectral independence improved the general upper bound to

$\varepsilon_i\in\{-1,1\}$7

where the tilde hides factors polynomial in $\varepsilon_i\in\{-1,1\}$8, while also proving the Beck–Fiala conjecture for sufficiently large column degree (Bansal et al., 5 Aug 2025). The method uses an SDP-guided discrete Brownian motion and imposes affine spectral-independence constraints to decouple the evolution of discrepancy across rows. A related exposition states the explicit form

$\varepsilon_i\in\{-1,1\}$9

and notes that this is asymptotically smaller than $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$0, thereby refuting Hajela’s conjecture that a $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$1 lower bound should hold (Bansal et al., 28 Aug 2026).

Several smoothed-analysis results establish much stronger discrepancy bounds after random perturbation. For Gaussian noise, if $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$2, where the entries of $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$3 are independent Gaussian variables of variance $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$4, then under

$$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$5

a suitable signing achieves discrepancy at most $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$6 with probability tending to one (Bansal et al., 2022). For Rademacher noise, if $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$7 has independent $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$8 entries and $$\left\|\sum_{i=1}^n \varepsilon_i v_i\right\|_\infty\le C.$$9, then

$A\in\mathbb R^{m\times n}$0

asymptotically almost surely (Aigner-Horev et al., 2023). These are smoothed statements and do not imply the worst-case conjecture for the unperturbed matrix.

The strongest explicit lower bound described in the supplied research is

$A\in\mathbb R^{m\times n}$1

where $A\in\mathbb R^{m\times n}$2 denotes the universal Komlós discrepancy constant. It is obtained from normalized matrices associated with unsatisfiable Boolean formulas and is optimal within the class arising from read-once resolution proofs (Kunisky, 2021). This lower bound does not contradict a constant upper bound; it shows only that any universal constant must be at least $A\in\mathbb R^{m\times n}$3.

4. Resolution of the conjecture

The paper “An elementary proof of the Komlós conjecture” proves the standard, non-prefix Komlós conjecture with the explicit universal constant $A\in\mathbb R^{m\times n}$4 (Karingula et al., 17 Sep 2026). Its theorem states that for every $A\in\mathbb R^{m\times n}$5 with $A\in\mathbb R^{m\times n}$6, there exist signs $A\in\mathbb R^{m\times n}$7 such that

$A\in\mathbb R^{m\times n}$8

The proof proceeds through two ingredients. First, it establishes a balancing lemma for a finitely supported probability distribution $A\in\mathbb R^{m\times n}$9. If $$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$0 is sufficiently close in total variation to each of its translates by $$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$1, specifically if

$$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$2

then signs can be chosen so that

$$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$3

belongs to the convex hull of the support of $$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$4.

Second, it constructs a mean-zero distribution supported on the cube $$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$5 satisfying

$$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$6

for every vector $$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$7 of Euclidean norm at most one. For rational vectors, the construction begins with the product density generated by the one-dimensional tent function

$$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$8

The density has controlled translation distance because its directional derivative satisfies

$$\operatorname{disc}(A) = \min_{x\in\{-1,1\}^n}\|Ax\|_\infty.$$9

A discretization argument transfers the continuous density to a finitely supported distribution on a rational grid. General real vectors are handled by approximation.

Applying the balancing lemma to $\|A_{*j}\|_2\le 1.$0 gives

$\|A_{*j}\|_2\le 1.$1

Multiplication by $\|A_{*j}\|_2\le 1.$2 yields

$\|A_{*j}\|_2\le 1.$3

The result is independent of both $\|A_{*j}\|_2\le 1.$4 and $\|A_{*j}\|_2\le 1.$5, and therefore establishes the conjecture.

The constant $\|A_{*j}\|_2\le 1.$6 is not claimed to be optimal. The proof is an elementary simplification of a preceding proof of Guo, Fang, and Lu, which obtained the sharper constant $\|A_{*j}\|_2\le 1.$7. The elementary argument is designed for conceptual transparency rather than quantitative optimization.

5. Proof mechanisms and variants

The splitting operator underlying the elementary proof is defined for a distribution $\|A_{*j}\|_2\le 1.$8 and a vector $\|A_{*j}\|_2\le 1.$9 by

$\operatorname{disc}(A)=O(1),$0

and

$\operatorname{disc}(A)=O(1),$1

The additional binary coordinate records whether both possible parents are available. The operation preserves total mass and contracts translation distance in the original coordinates.

When the operator is applied with $\operatorname{disc}(A)=O(1),$2, the two parents differ by $\operatorname{disc}(A)=O(1),$3. If $\operatorname{disc}(A)=O(1),$4, the mass of states with both parents is at least $\operatorname{disc}(A)=O(1),$5. This overlap provides enough freedom to choose the final sign while preserving the required mean. The inductive construction thus converts approximate translation invariance into an exact signing.

This mechanism differs from the SDP-guided stochastic-process proofs used for the earlier asymptotic bounds. Those methods evolve a fractional coloring $\operatorname{disc}(A)=O(1),$6, freeze coordinates approaching $\operatorname{disc}(A)=O(1),$7, and control row discrepancies through covariance constraints, regularized energies, and martingale inequalities. The elementary proof instead constructs a nearly translation-invariant distribution and uses convexity and total variation.

The result does not prove the strong Komlós conjecture, which asks for a single signing controlling every prefix: $\operatorname{disc}(A)=O(1),$8 The ordering of the vectors is fixed in this prefix formulation. The established theorem controls only the final sum.

6. Relaxations, applications, and current status

Complex discrepancy replaces signs by unit-modulus complex coefficients: $\operatorname{disc}(A)=O(1),$9 For real matrices, this is equivalent to rank-$m$0 vector discrepancy, where each sign is replaced by a unit vector in $m$1. A dimension-free bound

$m$2

and consequently

$m$3

was established for matrices with unit-bounded columns (Guillen et al., 14 Sep 2026). This resolves corresponding Gaussian-discrepancy formulations, but it does not by itself imply the Boolean Komlós conjecture.

The conjecture also has algorithmic consequences. Discrepancy bounds in the Komlós setting can be used in dynamic-programming algorithms for standard-form integer linear programs. If $m$4 denotes the worst Komlós discrepancy for matrices with $m$5 columns, then the resulting optimization and feasibility algorithms have running times controlled by $m$6 and $m$7, respectively, up to determinant and input-size factors (Gribanov et al., 10 Apr 2026). The proved bound $m$8 yields quasiexponential dependence on $m$9, while the Komlós conjecture would reduce this dependence to $v_1,\ldots,v_n\in\mathbb R^d$00.

The established status is therefore as follows:

• Classical Komlós conjecture: proved with the explicit universal bound $v_1,\ldots,v_n\in\mathbb R^d$01 (Karingula et al., 17 Sep 2026).
• Best constant: not determined; $v_1,\ldots,v_n\in\mathbb R^d$02 is a lower bound, while $v_1,\ldots,v_n\in\mathbb R^d$03 is an elementary explicit upper bound (Kunisky, 2021, Karingula et al., 17 Sep 2026).
• Strong prefix Komlós conjecture: remains open (Karingula et al., 17 Sep 2026).
• Algorithmic constant-bound signing: the elementary proof provides a finite construction for rational input vectors but does not establish a polynomial-time algorithm (Karingula et al., 17 Sep 2026).
• Complex and rank-$v_1,\ldots,v_n\in\mathbb R^d$04 relaxations: admit explicit dimension-free bounds (Guillen et al., 14 Sep 2026).
• Loebl–Komlós–Sós conjecture: a separate tree-containment conjecture; its approximate sparse form was proved through a four-paper series using sparse decompositions, regularized matchings, expansion structures, and tree-embedding configurations (Hladký et al., 2014, Hladký et al., 2014, Hladký et al., 2014, Hladký et al., 2014).