Author

Levi Segal

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Preprint Oct 2026

Improved Explicit and Algorithmic Lower Bounds for r(5,t)

We modify a construction of Brada\v{c} \cite{Bradac26} using a vertex ordering to give, for every prime power $q$, an explicit $K_5$ free graph with more than $q^7$ vertices and independence number below $600q^4$. This shows $$r(5,t)=\Omega(t^{7/4}).$$ Additionally, pruning a graph defined by the Coulter and Matthews p...

Amrit Kandasamy, Lawrence Zhou, Levi Segal · 0 citations

Preprint Sep 2026

Polynomial Length Cycles In Tough Graphs

We prove that every $n$-vertex $15$-tough graph contains a cycle of length $\Omega(n^{1/20})$.

Levi Segal, J. Verstraete · 2 citations · ⚡1

Preprint Sep 2026

Logarithmic Circumference In Tough Graphs

For every real $t>0$, we prove that every $2$-connected $t$-tough graph contains a cycle of length at least $\ell$ whenever $n \leq \ell(1 + 1/t)^{\lfloor \ell/2\rfloor - 1}$. This establishes the logarithmic bound conjectured by Broersma, van den Heuvel, Jung, and Veldman.

Levi Segal, J. Verstraete · 1 citation