Author

Swaroop G. Hegde

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Preprint Sep 2026

A somewhat sure note on an un-Schur problem

Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound...

Swaroop G. Hegde, Hitesh Kumar, Pratibha · 0 citations

Preprint Aug 2026

Refined upper bounds on Schur-like numbers

For positive integers $r, m$ and $N$, every $r$-coloring of $\{1, \dots, N\}$ contains a monochromatic solution to $x_1+\dots+x_{m+1}=y_1+\dots+y_m$ provided that $N \ge 3^r (r!)^{1/m}$, which is qualitatively optimal when $m$ is logarithmic in $r$.

Swaroop G. Hegde, Andrew Lott, G. Petridis et al. · 0 citations