A proof of the Ryser-Brualdi-Stein conjecture for large even $n$

TL;DR

The paper advances the Ryser-Brualdi-Stein conjecture by proving, for all sufficiently large even n, that every Latin square of order n contains a transversal of size n-1, and extends methods to Steiner triple systems to obtain matchings of size (n-4)/3. The authors develop a novel framework combining algebraic color properties, absorption, and addition structures within properly coloured, pseudorandom bipartite graphs, and deploy a semi-random nibble to construct large rainbow matchings that can be upgraded to perfect packings via absorption. This yields a robust, modular approach with potential to unlock further improvements for Latin arrays and related designs, and it demonstrates how algebraic structure and probabilistic methods can interplay to resolve longstanding combinatorial conjectures. The results significantly tighten prior bounds and confirm Brouwer’s conjecture for large Steiner triple systems, highlighting a deep connection between transversals, rainbow matchings, and algebraic colourings.

Abstract

A Latin square of order $n$ is an $n$ by $n$ grid filled using $n$ symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order $n$ contains a transversal with $n-1$ cells, and a transversal with $n$ cells if $n$ is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order $n$ has a transversal with $n-O(\log n/\log\log n)$ cells. Here, we show, for sufficiently large $n$, that every Latin square of order $n$ has a transversal with $n-1$ cells. We also apply our methods to show that, for sufficiently large $n$, every Steiner triple system of order $n$ has a matching containing at least $(n-4)/3$ edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with $n/3-O(\log n/\log\log n)$ edges, and proves a conjecture of Brouwer from 1981 for large $n$.

Figures (13)

• Figure 1: A $c,d$-colour-switcher of order 4 on the left with vertex set $\{w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2\}$ and colour set $\{c_1,c_2,c_3\}$ (see Definition \ref{['defn:colourswitcher']}), along with matchings $M_1$ and $M_2$ demonstrating its properties.
• Figure 2: An extremal colouring formed by taking disjoint equal-sized sets $A_0,A_1,B_0,B_1,C_0,C_1$ with equal odd order, and optimally colouring the complete bipartite graph with classes $A_0\cup A_1$ and $B_0\cup B_1$ so that, for each $i,j\in \mathbb{Z}_2$, edges between $A_i$ and $B_j$ have colour in $C_{i+j}$.
• Figure 3: An $e_1,e_2$-edge-switcher of order 4 depicted on the left with vertex set $\hat{V}=\{w_1,x_1,w_2,x_2\}$ and colour set $\hat{C}=\{d,d',c'\}$ (see Definition \ref{['defn:edgeexchanger']}), along with two exactly-$\{d,d',c'\}$-rainbow matchings with vertex set $\hat{V}\cup \{u_2,v_2\}$ and $\hat{V}\cup \{u_1,v_1\}$, respectively.
• Figure 4: An absorber on the left with vertex set $\{w,x\}\cup (\cup_{i\in [100]}\hat{V}_i\cup\{w_i,x_i\})$ and colour set $\{d,d'\}\cup(\cup_{i\in [100]}\hat{C}_i)$ (see Definition \ref{['defn:Eabs']}), which can absorb the two vertices in $\{u_i,v_i\}$ for any one $i\in [100]$, and incorporates the $wx,w_ix_i$-switchers $(\hat{V}_i,\hat{C}_i)$, $i\in [100]$. On the right, $u_2$ and $v_2$ are absorbed by turning $(\hat{V}_2,\hat{C}_2)$ to cover $x$ and $w$, turning each other switch $(\hat{V}_i,\hat{C}_i)$ to cover $x_i$ and $w_i$, and using the exactly-$\{d,d'\}$-rainbow matching $\{u_2w_2,v_2x_2\}$ to cover $u_2,v_2,w_2$ and $x_2$.
• Figure 5: The initial partition of $V(G)$ and $C(G)$ at i), and the subsequent development of these partitions as the matchings $M_1$ to $M_4$ are found, where one colour and two vertices in $G$ are not used.

Theorems & Definitions (128)

• Conjecture 1.1: The Ryser-Brualdi-Stein conjecture
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5: Brouwer
• Theorem 1.6
• Lemma 2.1: montgomery2018spanning
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3: Almost-perfect rainbow matchings