Author

Jesse T. Geneson

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Preprint Sep 2026

Deletion thresholds and exponential examples for complete sequences

We prove that the pairs of integers $0\le m<n$ for which a nondecreasing integer sequence can remain complete after every deletion of $m$ terms and become incomplete after every deletion of $n$ terms are exactly those with $m\le1$. Here a sequence is complete if every sufficiently large integer is a finite sum of terms...

Jesse T. Geneson · 0 citations

Jul 2026

Linear extremal bounds for a family of forbidden 0-1 matrices

Fulek defined the $0$-$1$ matrix \[ L_3=\begin{pmatrix} 1&0&0&1&0\\ 0&0&0&0&1\\ 0&1&1&0&0 \end{pmatrix} \] and asked whether $\text{ex}(n,L_3) = O(n)$. We prove that every $r\times s$ $0$-$1$ matrix avoiding $L_3$ has at most $27r+2s$ $1$ entries. Fulek's general lower bound construction has $6n-8$ $1$ entries, so \[ 6...

Jesse T. Geneson · arXiv.org · 0 citations

Preprint Aug 2026

Enumerating forcing and strongly forcing (0,1)-matrices

Let $Q$ be a nonzero $s\times t$ $(0,1)$-pattern, and let $m\ge s$ and $n\ge t$. An $m\times n$ matrix is strongly $Q$-forcing if every $1$-entry belongs to an $s\times t$ submatrix equal to $Q$. Let $F^{*}(m,n,Q)$ count these matrices. Put $H=m-s+1$ and $W=n-t+1$. We prove \[ F^{*}(m,n,Q)\ge 2^{HW}. \] Writing $r$ and...

Lei Cao, Jesse T. Geneson · 0 citations