If you find a mistake, omission, etc., please let me know by e-mail.

Apropos of mistakes etc., here’s a short list of corrections to the textbook from previous iterations of the course (formerly Math 155r).

Also, the existence of nontrivial Steiner systems with t ≥ 6, described on the bottom of page 2 as “possibly the most important open problem in design theory”, was finally solved in 2014 by Peter Keevash.

The orange balls mark our current location in the course, and the current problem set.

“faculty legislation requires all instructors to include a statement outlining their policies regarding collaboration on their syllabi” — as stated in h1.pdf: for homework, “As usual in our department, you are allowed — indeed encouraged — to collaborate on solving homework problems, but must write up your own solutions.” For the final project or presentation, work on your own even if another student has chosen the same topic. (As with theses etc. it is still OK to ask peers to read drafts of your paper, or see dry runs of your presentation, and make comments.) In all cases, acknowledge sources as usual, including peers in your homework group.

h1.pdf: Ceci n’est pas un Math 165 syllabus.

h2.pdf: Handout #2, containing some basic definitions and facts about finite fields.

h3.pdf: Handout #3: outline of a proof of the simplicity of ${\rm PSL}_2(F)$ ($F$ a finite field of at least 4 elements) and ${\rm PSL}_n(F)$ for $n \geq 3$ and any finite field $F$.

h4.pdf: Handout #4: The exceptional isomorphism ${\rm PSL}_2({\bf F}_7) \to {\rm GL}_3({\bf F}_7)$ via the automorphism group of the 3-(/,4,1) Steiner system

Here are Andries E. Brouwer’s tables of strongly regular graphs. For instance, the first table shows all parameters for $v \leq 50$ allowed by the integrality condition. Green means the graph exists, in which case the first column has “!” if it is unique up to automorphisms, and “$n$rdquo; with some $n \geq 1$ if the number of isomorphism classes is known to be exactly $n$ (see the comments column: if there are no comments, look at the entry above for the complementary-graph parameters). Red means there is no such graph (and the comments indicate why not). Yellow means that existence is an open question; there is no such case for $v \leq 50,$ but the next page (for $50 \leq v \leq 100$) already shows a few examples.

January 28: Introduction: basic definitions and questions [Fraktur $\mathfrak D$, $\mathfrak B$ correspond to the textbook’s script D and B, which are normally \cal D, \cal B in TeX, but this might not be recognized by MathJax.]

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p1.pdf: First problem set, exploring the Fano plane (and generalizations) and Petersen graph from the introductory handout.

The use of English words to encode combinatorial structure, as in {BUD, BYE, DOE, DRY, ORB, RUE, YOU} $\cong \Pi_2$, is one of many bits of mathematics (and wordplay) that I was introduced to by the writings of the late great Martin Gardner. In page 208 of Mathematical Carnival (New York: Knopf, 1975) he introduces the following game: Each of the following words is printed on a card: HOT, HEAR, TIED, FORM, WASP, BRIM, TANK, SHIP, WOES. The nine cards are placed face up on the table. Players take turns removing a card. The first to hold three cards that bear the same letter is the winner. (The Canadian mathematician Leo Moser , who devised this game, called it “Hot.”) What familiar combinatorial structure does this set of words encode? Hint: “Hot” is the last of three games described on this page; the first is: Nine playing cards, with values from ace[=1] to nine, are face up on the table. Players take turns picking a card. The first to obtain three cards that add to 15 is the winner. (The endnotes for this chapter “16. Jam, Hot, and other games” cite Leo Moser, “The Game is Hot.” Recreational Mathematics Magazine, Vol.1, June, 1961, pages 23–24.) Another variation, using the words BET, BUG, CLOG, EACH, FRAUD, GEM, LAMB, MUTT, STILL, used to be here.