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arXiv · 2608.06440

An infinite family of doubly saturated $R(3,t)$-good graphs

Abhishek Saigal·Akaash R. Parthasarathy
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Abstract

For every odd integer $t\ge17$, we prove that an explicit circulant graph on $5t-10$ vertices is doubly saturated $R(3,t)$-good. The graph is triangle-free and has independence number $t-1$. Adding any nonedge creates a triangle, whereas deleting any edge creates an independent set of order $t$. This settles Conjecture 2 of Przybocki, Mackey, Heule, and Subercaseaux. A cyclic sumset identity and explicit witnesses prove the local saturation properties. Writing $t=2m+1$, a five-layer reduction proves the independence bound via a uniform affine certificate for $m\ge30$ and an exhaustive checker for $8\le m\le29$. The checker soundness and the complete argument are formalized in Lean 4.32.2. Consequently, $2t-1\le \operatorname{DS}(3,t)\le 5t-10$ for odd $t\ge17$.

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Abhishek Saigal, Akaash R. Parthasarathy. 2026-08-06. An infinite family of doubly saturated $R(3,t)$-good graphs. https://arxiv.org/abs/2608.06440

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