Hunter Snevily

Hunter received his undergraduate degree from Emory University in 1981,^[1] and his Ph.D. degree from the University of Illinois Urbana-Champaign under the supervision of Douglas West in 1991.^[1][3] After a postdoctoral fellowship at Caltech, where he mentored many students, Hunter took a faculty position at the University of Idaho in 1993 where he was a professor until 2010.^[1] He retired early ^[4] while fighting with Parkinsons,^[1][2] but continued research in mathematics till his last days.^[1][2]

The following are some of Hunter's most important contributions (as discussed in ^[2]):

• Hunter formulated a conjecture (1991) ^[5] bounding the size of a family of sets under intersection constraints. He conjectured that if ${\displaystyle {\mathcal {L}}}$ is a set of ${\displaystyle k}$ positive integers and ${\displaystyle \{A_{1},A_{2},\ldots ,A_{m}\}}$ is a family of subsets of an ${\displaystyle n}$-set satisfying ${\displaystyle |A_{i}\cap A_{j}|\in {\mathcal {L}}}$ whenever ${\displaystyle i\neq j}$, then ${\displaystyle m\leq \sum _{i=0}^{k}{n-1 \choose i}}$. His conjecture was ambitious in a way it would beautifully unify classical results of Nicolaas Govert de Bruijn and Paul Erdős (1948),^[6] Bose (1949),^[7] Majumdar (1953),^[8] H. J. Ryser (1968),^[9] Frankl and Füredi (1981),^[10] and Frankl and Wilson (1981).^[11] Hunter finally proved his conjecture in 2003^[12]
• Hunter made important contribution to the well known Chvátal's Conjecture (1974)^[13] which states that every hereditary family ${\displaystyle {\mathcal {F}}}$ of sets has a largest intersecting subfamily consisting of sets with a common element. Schönheim^[14] proved this when the maximal members of ${\displaystyle {\mathcal {F}}}$ have a common element. Vašek Chvátal proved it when there is a linear order on the elements such that ${\displaystyle \{b_{1},b_{2},\ldots ,b_{k}\}\in {\mathcal {F}}}$ implies ${\displaystyle \{a_{1},a_{2},\ldots ,a_{k}\}\in {\mathcal {F}}}$ when ${\displaystyle a_{i}\leq b_{i}}$ for ${\displaystyle 1\leq i\leq k}$. A family ${\displaystyle {\mathcal {F}}}$ has ${\displaystyle x}$ as a dominant element if substituting ${\displaystyle x}$ for any element of a member of ${\displaystyle {\mathcal {F}}}$ not containing ${\displaystyle x}$ yields another member of ${\displaystyle {\mathcal {F}}}$. Hunter's 1992 result^[15] greatly strengthened both Schönheim's result and Chvátal's result by proving the conjecture for all families having a dominant element; it was major progress on the problem.
• One of his most cited papers^[16] is with Lior Pachter and Bill Voxman^[17] on Graph pebbling. This paper and Hunter's later paper^[18] with Foster added several conjectures on the subject and together have been cited in more than 50 papers.
• Hunter made important contributions^[19][20][21] on the Snake-in-the-box problem and on the Graceful labeling of graphs.
• One of Hunter's conjectures (1999)^[22] became known as Snevily's Conjecture:^[23] Given an abelian group ${\displaystyle G}$ of odd order, and subsets ${\displaystyle \{a_{1},a_{2},\ldots ,a_{k}\}}$ and ${\displaystyle \{b_{1},b_{2},\ldots ,b_{k}\}}$ of ${\displaystyle G}$, there exists a permutation ${\displaystyle \pi }$ of ${\displaystyle [k]}$ such that ${\displaystyle a_{1}+b_{\pi }(1),a_{2}+b_{\pi }(2),\ldots ,a_{k}+b_{\pi }(k)}$ are distinct. Noga Alon^[24] proved this for cyclic groups of prime order. Dasgupta et al. (2001).^[25] proved it for all cyclic groups. Finally, after a decade, the conjecture was proved for all groups by a young mathematician Arsovski.^[26] Terence Tao devoted a section to Snevily's Conjecture in his well-known book Additive Combinatorics.
• Hunter collaborated the most^{[27][28][29][21][30][31][32][33]} with his long-term friend^[2] André Kézdy. After retirement, he became friends with Tanbir Ahmed^[2] and explored experimental mathematics that resulted in several publications ^{[34][35][36][37][38][39]}