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Adam Zsolt Wagner

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math.CO Artificial Intelligence Human-Computer Interaction

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preprint2026arXiv

Intentmaking and Sensemaking: Human Interaction with AI-Guided Mathematical Discovery

Artificial intelligence offers powerful new tools for scientific discovery, but the interaction paradigms required to effectively harness these systems remain underexplored. In this paper, we present findings from a formative user study with 11 expert mathematicians who used AlphaEvolve, an evolutionary coding agent, to tackle advanced problems in their fields of expertise. We identify and characterize a distinct workflow we term intentmaking, the iterative process of discovering, defining, and refining one's experimental goals through active system interaction. We frame this as a natural extension to sensemaking, the cognitive process of building an understanding of complex or novel data. We suggest that users enter a cycle of intentmaking (defining and updating their experiment) and sensemaking (interpreting the results) which repeats many times during the course of an investigation. Our documentation of these themes suggests an approach to designing AI tools for scientific discovery that goes beyond the existing question/answer model of many current systems, treating them as collaborative instruments rather than opaque black-box assistants.

preprint2020arXiv

Families in posets minimizing the number of comparable pairs

Given a poset $P$ we say a family $\mathcal{F}\subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$ , among all families of size $M$ in $P$ , centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$ . We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbb{F}_q^n$ has the centeredness property. Several open questions are also given.

preprint2020arXiv

Infinite Sperner's theorem

One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice $2^{[n]}$ has size $Θ\big(\frac{2^n}{\sqrt{n}}\big)$ . Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that infinite antichains should be "thinner" than the corresponding finite ones. More precisely, if $\mathcal{F}\subset 2^{\mathbb{N}}$ is an antichain, then $\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log n}\right)^{-1}=0.$ Our main result shows that this bound is essentially tight, that is, we construct an antichain $\mathcal{F}$ such that $\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log^{C} n}\right)^{-1}>0$ holds for some absolute constant $C>0$ .

preprint2018arXiv

Partition problems in high dimensional boxes

Alon, Bohman, Holzman and Kleitman proved that any partition of a $d$ -dimensional discrete box into proper sub-boxes must consist of at least $2^d$ sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a $d$ -dimensional box of odd size, and they asked whether the trivial construction consisting of $3^d$ boxes is best possible. We show that approximately $2.93^d$ boxes are enough, and consider some natural generalisations.

preprint2016arXiv

Further applications of the Container Method

Recently, Balogh--Morris--Samotij and Saxton--Thomason proved that hypergraphs satisfying some natural conditions have only few independent sets. Their main results already have several applications. However, the methods of proving these theorems are even more far reaching. The general idea is to describe some family of events, whose cardinality a priori could be large, only with a few certificates. Here, we show some applications of the methods, including counting $C_4$ -free graphs, considering the size of a maximum $C_4$ -free subgraph of a random graph and counting metric spaces with a given number of points. Additionally, we discuss some connections with the Szemerédi Regularity Lemma.

preprint2016arXiv

Kleitman's conjecture about families of given size minimizing the number of $k$ -chains

A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\mathcal{F}\subseteq \mathcal{P}(n)$ that does not contain a $2$ -chain $F_1\subsetneq F_2$ . Erdős later extended this result and determined the largest family not containing a $k$ -chain $F_1\subsetneq \ldots \subsetneq F_k$ . Erdős and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result. This question was resolved for $2$ -chains by Kleitman in $1966$ , who showed that amongst families of size $M$ in $\mathcal{P}(n)$ , the number of $2$ -chains is minimized by a family whose sets are taken as close to the middle layer as possible. He also conjectured that the same conclusion should hold for all $k$ , not just $2$ . The best result on this question is due to Das, Gan and Sudakov who showed that Kleitman's conjecture holds for families whose size is at most the size of the $k+1$ middle layers of $\mathcal{P}(n)$ , provided $k\leq n-6$ . Our main result is that for every fixed $k$ and $ε>0$ , if $n$ is sufficiently large then Kleitman's conjecture holds for families of size at most $(1-ε)2^n$ , thereby establishing Kleitman's conjecture asymptotically. Our proof is based on ideas of Kleitman and Das, Gan and Sudakov. Several open problems are also given.

preprint2016arXiv

Large subgraphs in rainbow-triangle free colorings

Fox--Grinshpun--Pach showed that every $3$ -coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $Ω\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such coloring contains a $2$ -colored subgraph with chromatic number at least $n^{2/3}$ , and this is best possible. We further show that for fixed positive integers $s,r$ with $s\leq r$ , every $r$ -coloring of the edges of the complete graph on $n$ vertices without a rainbow triangle contains a subgraph that uses at most $s$ colors and has chromatic number at least $n^{s/r}$ , and this is best possible. Fox--Grinshpun--Pach previously showed a clique version of this result. As a direct corollary of our result we obtain a generalisation of the celebrated theorem of Erdős-Szekeres, which states that any sequence of $n$ numbers contains a monotone subsequence of length at least $\sqrt{n}$ . We prove that if an $r$ -coloring of the edges of an $n$ -vertex tournament does not contain a rainbow triangle then there is an $s$ -colored directed path on $n^{s/r}$ vertices, which is best possible. This gives a partial answer to a question of Loh.

preprint2016arXiv

On the number of union-free families

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most $(1+o(1))\binom{n}{n/2}$ . Later, Burosch--Demetrovics-Katona-Kleitman-Sapozhenko asked for the number $α(n)$ of such families, and they proved that $2^{\binom{n}{n/2}}\leq α(n) \leq 2^{2\sqrt{2}\binom{n}{n/2}(1+o(1))}$ . They conjectured that the constant $2\sqrt{2}$ can be removed in the exponent of the right hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.

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Adam Zsolt Wagner

Research overview

Is this your work?

8 publications

Intentmaking and Sensemaking: Human Interaction with AI-Guided Mathematical Discovery

Families in posets minimizing the number of comparable pairs

Infinite Sperner's theorem

Partition problems in high dimensional boxes

Further applications of the Container Method

Kleitman's conjecture about families of given size minimizing the number of $k$ -chains

Large subgraphs in rainbow-triangle free colorings

On the number of union-free families

See the researcher in context

Building this map preview

Adam Zsolt Wagner

Research overview

Is this your work?

8 publications

Intentmaking and Sensemaking: Human Interaction with AI-Guided Mathematical Discovery

Families in posets minimizing the number of comparable pairs

Infinite Sperner's theorem

Partition problems in high dimensional boxes

Further applications of the Container Method

Kleitman's conjecture about families of given size minimizing the number of kkk-chains

Large subgraphs in rainbow-triangle free colorings

On the number of union-free families

See the researcher in context

Building this map preview

Kleitman's conjecture about families of given size minimizing the number of $k$ -chains