New bounds for Ryser’s conjecture and related problems

A Latin square of order n n is an alttext="n times n"> × encoding="application/x-tex">n \times n array filled with symbols such that each symbol appears only once in every row or column and a transversal collection cells which do not share the same row, symbol. The study squares goes back more than 200 years to work Euler. One most famous open problems this area conjecture Ryser-Brualdi-Stein from 60s says encoding="application/x-tex">n\times contains minus 1"> − encoding="application/x-tex">n^{2-\varepsilon } proofs combine novel way semi-random method together robust expansion properties edge-coloured pseudorandom graphs rainbow covering all but encoding="application/x-tex">O(\log n/\log vertices. All previous results, based method, left uncovered at least alttext="normal Omega alpha Baseline Ω<!-- Ω <mml:mi>α<!-- α encoding="application/x-tex">\Omega (n^{\alpha }) (for some constant alttext="alpha greater-than 0"> &gt; 0 encoding="application/x-tex">\alpha &gt;0 )