Croatian mathematicians solve problems that have baffled experts for 40 years

Croatian mathematicians solve problems that have baffled experts for 40 years

Two Croatian mathematicians from the University of Zagreb’s Faculty of Science, have achieved an incredible success by solving two famous Erdős problems, becoming the only mathematicians currently living and working in Croatia who have accomplished this feat.

Paul Erdős (1913-1996), the renowned Hungarian mathematician, posed a series of problems with elementary and straightforward formulations that proved to be unsolvable for established mathematicians. 

The resolution of these challenges demanded innovation and new techniques, attracting the attention of mathematicians worldwide.

The Erdős Problems

Erdős, a maverick, presented problems spanning discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. 

These seemingly simple problems eluded solutions from the mathematical community, showcasing Erdős’s penchant for pushing the boundaries of conventional understanding.

The Achievement of Croatian Mathematicians

Enter Croatian mathematicians, Professor Vjekoslav Kovač, and assistant Adrian Beker from the Mathematical Department of the Faculty of Science in Zagreb (PMF). 

They successfully cracked two famous Erdős problems highlighted on the “Erdős problems” website curated by British mathematician Thomas Bloom, the faculty announced. 

Professor Vjekoslav Kovač (Photo: Croatian Mathematical Society/PMF)

Recently, Kovač solved problem #189, demonstrating the existence of a plane colouring with finitely many colours such that there is no unit-square with all four vertices of the same colour. Subsequently, Beker solved problem #356, showing that for every natural number n, there exist natural numbers a1<a2<…, PMF and the Croatian Mathematical Society said in a statement.

Adrian Beker (Photo: Croatian Mathematical Society/PMF)

Significant accomplishment

As explained by PMF, Paul Erdős was one of the most renowned mathematicians of the last century. He earned this title mainly by posing open problems with elementary and simple formulations, which were just beyond the reach of known mathematics, requiring imaginative solutions or the development of new techniques.

It is impossible to gather all the problems Erdős posed during his lifetime, but British mathematician Thomas Bloom initiated a project to collect a large number of Erdős’s most interesting problems. Currently, the Erdős problems website features about 500 problems, with only about a fifth of them solved to date.

As emphasised by the faculty, Kovač and Beker have published their proofs so far only in the form of preprints, but the solutions themselves are very elegant and have already been verified and confirmed on the ‘Erdős problems’ website.

Among Croatian mathematicians, in terms of origin, the late Branko Grünbaum is also mentioned as one of the solvers, according to the statement from the Mathematical Department of PMF.