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Let $S=\{1,2,\ldots,n\}$. Let $A_i\subset S$ for $i\in\{1,2,\ldots,m\}$. Impose the following conditions

  • $|A_i|=r$ with $r<n$ for all $i$.
  • $|A_i\cap A_j|=t$ for all $i\neq j$, with $t<r$.

Let $n,r,t$ be fixed. What is the maximum number $m$ of subsets $A_i$ that satisfy these conditions?

The question arises from my observation of a game called Rafly, where there are 55 cards, each with 8 images on them. No matter which 2 cards you pick, they have one and only one common image. I want to generalize this game by using the minimum number of $n$ images, having $m$ cards, each with $r$ images and $t$ common images. However, I am having trouble with finding $m$ given $n,r$ and $t$.

In the case of $t=1$ I start to build the cards like this:

$$A_1 = \{1,2,\ldots,r\}.$$ A new card must have one common element with the first card: $$A_2 = \{c_{1,2}, r+1,\ldots, 2r-1 \},$$ where $c_{1,2}\in A_1$. A new card must have one common element with the second and first cards: $$A_3 = \{c_{1,3},c_{2,3}, 2r,2r+1,\ldots, 3r-3\},$$ where $c_{1,3}\in A_1$ and $c_{2,3}\in A_2\setminus A_1$. Following this reasoning I conclude that there are $n=r(r+1)/2$ different images. But how many cards are there? How are these quantities related to $t$? Thanks a lot.

bof
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Vladimir Vargas
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    There is a problem in your reasoning: What if $c_{1,3}=c_{2,3}$? More generally, you can have a "star-shaped" collection of sets all with a single common element and a bunch of unique elements, and there is no bound on how many you can have, meaning that for $r,t$ fixed there is no maximum $m$, assuming $n$ is allowed to grow as well. – Mario Carneiro Jan 05 '19 at 01:18
  • @MarioCarneiro Right, I corrected the errors – Vladimir Vargas Jan 05 '19 at 01:27
  • Well no, the error is still there, because $c_{1,3}=c_{2,3}$ is a legal choice. You don't know that they are distinct, and so you don't know that the image count is up to $3r-3$ after the third card - it could also be $3r-2$. But I guess you mean to say that there are at least $3r-3$ images in use in the first three cards. You can continue this lower bound up until you get to $A_r$, at which point you have used $r(r+1)/2$ images and $m=r$, but then the lower bound stops and $m$ can possibly keep increasing, reusing all the old cards for a while. The true upper bound for $m$ is farther on. – Mario Carneiro Jan 05 '19 at 01:51
  • @MarioCarneiro if $c_{2,3}\in A_2\setminus A_1$ it can't be $c_{1,3}$ – Vladimir Vargas Jan 05 '19 at 02:07
  • Yes, I know you said that, but it's not true. You say "a new card must have one common element with the second and first cards" but it does not follow that there are two distinct common elements; you could have one element common to all three cards. – Mario Carneiro Jan 05 '19 at 02:10
  • @MarioCarneiro Now I get it, yes. I am doing that in order to minimize the number $n$ of images in the game. I'm not expressing myself very formally, because $n$ is fixed a priori. However I do want to have as few images as possible. – Vladimir Vargas Jan 05 '19 at 02:11
  • By the way, in Rafly is it the case that every image is on the same number of cards, and if so what number is it? – Mario Carneiro Jan 05 '19 at 02:13
  • @MarioCarneiro I don't know. I played it yesterday night but I am not the owner. There is no much info on the internet either – Vladimir Vargas Jan 05 '19 at 02:17
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    This game is known as "Spot It!" in the US and "Dobble" in Europe. Searching for these names turns up several questions about these games across StackExchange. For example, on this site: one, two, three. You might find (partial) answers there or by following further links. – Alex Kruckman Jan 05 '19 at 02:19
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    Also, this is off-topic, but what's the deal with your profile picture? – Alex Kruckman Jan 05 '19 at 02:20
  • I knew my countrymen couldn't come up with something this creative! XD My profile picture depicts the voluptuous beauty of Euler's identity – Vladimir Vargas Jan 05 '19 at 02:23
  • I found this: very insightful https://www.tandfonline.com/doi/abs/10.4169/mathhorizons.22.4.8?journalCode=umho20 – Vladimir Vargas Jan 05 '19 at 02:43

1 Answers1

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Your problem in the $t=1$ case was asked here. I will provide a generalization of those ideas to your problem.

There are $\binom{n}{t+1}$ subsets of $S$ with size $t+1$. Each set $A_i$ has $\binom{r}{t+1}$ subsets of size $t+1$. Furthermore, the $\binom{r}{t+1}$ subsets of size $t+1$ for $A_i$ must be different then those for $A_j$, because $A_i$ and $A_j$ only have $t$ elements in common. Therefore, counting up the total number of subsets of size $t+1$ found in some set $A_i$, we get $$ m\cdot\binom{r}{t+1}\le \binom{n}{t+1} $$ This give you an upper bound on $m$.

If equality is attained, this would mean that every subset of size $t+1$ is contained in exactly one set $A_i$. . This means that you have a Steiner system $S(t+1,r,n)$. Not much is known about when Steiner systems exist; see the other answer for some more information.

Mike Earnest
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